Q2 Expand products and squares of binomials | Ontario Grade 10 Mathematics

Ontario Grade 10 Mathematics

Quadratic Relations

Building Trinomials from Two Brackets

You already know how to multiply a monomial by a bracket — that is Grade 9 work. This lesson takes the next step: multiplying a bracket by a bracket. The result is usually a trinomial (three terms), and being able to write that expanded form quickly and accurately is a skill used throughout the rest of this course whenever you work with quadratic expressions. We will build the idea from a concrete area picture, move to a step-by-step algebra method, and then look at the special case of squaring a binomial, where a pattern makes the work faster.

What you will learn

Multiply two different binomials together using the distributive property and collect like terms.
Expand a binomial squared by recognising it as a product of two identical factors.
Identify and avoid common sign and middle-term errors when expanding.
Connect the algebra to an area model to check your reasoning.

Grade 9 Bridge: The Distributive Property

Before multiplying two brackets together, recall what distributing means. When you see

3(x + 4)

, you multiply the

3

by every term inside:

3 \cdot x + 3 \cdot 4 = 3x + 12

. Each term inside the bracket gets its own copy of the factor outside.

Expanding two brackets works the same way, just twice. Think of the first bracket as the outside factor. Every term in the first bracket must multiply every term in the second bracket. Nothing gets left out.

A helpful memory tool is the word FOIL, which stands for First, Outer, Inner, Last. This is simply a checklist of the four multiplications you must do when each bracket has exactly two terms. We will use it as a checklist, not a magic rule.

Distributing means every term outside multiplies every term inside a bracket.
With two binomials, there are exactly four individual multiplications to perform.
FOIL (First, Outer, Inner, Last) is a checklist to keep those four multiplications organised.
After multiplying, collect any like terms to simplify.

Expanding a Product of Two Binomials

Consider the general product

(ax + b)(cx + d)

. To expand it, take each term of the first bracket and multiply it by each term of the second bracket. That gives four partial products.

First: multiply the first terms together,

ax \cdot cx = acx^2

. Outer: multiply the outer terms,

ax \cdot d = adx

. Inner: multiply the inner terms,

b \cdot cx = bcx

. Last: multiply the last terms,

b \cdot d = bd

.

Writing all four products in a row gives

acx^2 + adx + bcx + bd

. The two middle terms,

adx

and

bcx

, are like terms because they both contain

x

to the first power. You combine them by adding their coefficients, giving the final trinomial

acx^2 + (ad + bc)x + bd

.

Always watch the signs. If a term is negative, that negative sign travels with it into every multiplication involving that term. Treating a subtraction as adding a negative number helps avoid mistakes.

(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd

Four multiplications: First, Outer, Inner, Last.
The outer and inner products are like terms and must be added together.
Carry negative signs into every multiplication — they belong to their term.
The final result of expanding two binomials is usually a trinomial.

Area Model: Seeing Why It Works

An area model turns the algebra into a picture. Imagine a rectangle whose width is

(x + 3)

and whose height is

(x + 5)

. The total area equals width times height, so it equals

(x + 3)(x + 5)

.

Split the rectangle into four smaller pieces by drawing one vertical line after the

x

part of the width, and one horizontal line after the

x

part of the height. The top-left piece has area

x \cdot x = x^2

. The top-right piece has area

3 \cdot x = 3x

. The bottom-left piece has area

x \cdot 5 = 5x

. The bottom-right piece has area

3 \cdot 5 = 15

.

Adding all four pieces:

x^2 + 3x + 5x + 15 = x^2 + 8x + 15

. This matches what FOIL gives, which confirms the algebra is correct. Use this picture any time you feel uncertain about whether you have the right number of terms.

A rectangle split into four parts models the four multiplications in FOIL.
Each small rectangle's area corresponds to one of the four partial products.
The total area of the big rectangle equals the expanded trinomial.
This visual check is especially useful when signs or coefficients look surprising.

Squaring a Binomial: A Special Case

Squaring a binomial means multiplying it by itself. For example,

(x + 4)^2

means

(x + 4)(x + 4)

. It is not the same as

x^2 + 4^2

, which is one of the most common errors in this unit. Write the squared binomial as two separate, identical brackets first, and then expand.

Applying FOIL to

(x + 4)(x + 4)

: First gives

x^2

, Outer gives

4x

, Inner gives

4x

, Last gives

16

. Combining:

x^2 + 4x + 4x + 16 = x^2 + 8x + 16

.

Notice the pattern: the first term squared, plus twice the product of the two terms, plus the last term squared. In symbols,

(a + b)^2 = a^2 + 2ab + b^2

. You can use this pattern as a shortcut once you understand why it works — but the understanding comes from doing FOIL first.

The same pattern applies when the sign is negative:

(a - b)^2 = a^2 - 2ab + b^2

. The middle term becomes negative, but the last term is still positive because a negative times a negative is positive.

(a + b)^2 = a^2 + 2ab + b^2

$(x + a)^2$ means $(x + a)(x + a)$ , not $x^2 + a^2$ .
FOIL on identical brackets always produces a middle term of $2ab$ .
Pattern: $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ .
The last term of a squared binomial is always positive regardless of the sign between terms.

Putting It All Together: Strategy and Checks

When you face any expansion problem, follow a consistent routine. First, write both brackets side by side if a squared notation hides one of them. Second, identify each of the four FOIL multiplications and write the partial products before collecting anything. Third, circle or underline the like terms, then add their coefficients. Fourth, write the simplified trinomial and double-check the sign of every term.

A quick numerical check is useful: choose a simple value like

x = 2

, evaluate the original brackets on one side, and evaluate your expanded trinomial on the other. If both sides give the same number, your expansion is almost certainly correct. If they differ, a sign or arithmetic error is hiding somewhere.

Consistent use of these steps reduces errors even with messier coefficients, negative terms, or leading coefficients other than 1. The process is always the same; only the numbers change.

Always rewrite a squared binomial as two brackets before expanding.
Write all four partial products first, then collect like terms in a second step.
Substitute a test value into both forms to verify your answer quickly.
The middle coefficient is the sum of the outer and inner products — check it separately if unsure.

Area Model for (x + 3)(x + 5)

x	5
x	x²	5x
3	3x	15

FOIL Checklist at a Glance

Step	Terms Multiplied	What It Contributes
First	First term × First term	The x² term
Outer	First term × Last term of 2nd bracket	Part of the x term
Inner	Last term of 1st bracket × First term of 2nd bracket	Part of the x term
Last	Last term × Last term	The constant term

Worked example

Expanding a Product of Two Different Binomials

Expand and simplify

(2x + 3)(x - 5)

.

Identify the four FOIL multiplications
Label the terms: First terms are $2x$ and $x$ ; Outer terms are $2x$ and $-5$ ; Inner terms are $3$ and $x$ ; Last terms are $3$ and $-5$ . Writing them out as a list before combining prevents missed terms.
Multiply the First terms
Multiply $2x$ by $x$ . When you multiply two powers of $x$ , you add the exponents: $x^1 \cdot x^1 = x^2$ .
$2x \ cdot x = 2x^2$
Multiply the Outer terms
Multiply $2x$ by $-5$ . The negative sign belongs to $-5$ , so the product is negative.
$2x · (-5) = -10x$
Multiply the Inner terms
Multiply $3$ by $x$ . Both are positive, so the product is positive.
$3 · x = 3x$
Multiply the Last terms
Multiply $3$ by $-5$ . Positive times negative gives a negative product.
$3 · (-5) = -15$
Write all four partial products in a row
Place the four results side by side. This is not the final answer yet because two like terms still need to be combined.
$2x^2 - 10x + 3x - 15$
Collect the like terms
The terms $-10x$ and $+3x$ are like terms because they both contain $x$ to the first power. Add their coefficients: $-10 + 3 = -7$ .
$2x^2 - 7x - 15$

Answer:

(2x + 3)(x - 5) = 2x^2 - 7x - 15

Check: Substitute

x = 1

. Left side:

(2(1) + 3)(1 - 5) = (5)(-4) = -20

. Right side:

2(1)^2 - 7(1) - 15 = 2 - 7 - 15 = -20

. Both sides match, so the expansion is correct.

Worked example

Expanding a Squared Binomial

Expand and simplify

(3x - 4)^2

.

Rewrite the square as two identical brackets
The exponent 2 means the bracket is multiplied by itself. Writing it as two separate brackets makes it clear that four multiplications are needed, and prevents the mistake of just squaring each term individually.
$(3x - 4)(3x - 4)$
Multiply the First terms
Multiply $3x$ by $3x$ . Multiply the coefficients $3 \times 3 = 9$ , then multiply the variable parts $x \cdot x = x^2$ .
$3x · 3x = 9x^2$
Multiply the Outer terms
Multiply $3x$ by $-4$ . Positive times negative gives a negative result.
$3x · (-4) = -12x$
Multiply the Inner terms
Multiply $-4$ by $3x$ . Negative times positive gives a negative result. Notice this is the same as the outer product — that always happens when both brackets are identical.
$(-4) · 3x = -12x$
Multiply the Last terms
Multiply $-4$ by $-4$ . Negative times negative gives a positive result. This confirms that the constant term of a squared binomial is always positive.
$(-4)(-4) = 16$
Write all four partial products in a row
Place all four results together before simplifying.
$9x^2 - 12x - 12x + 16$
Collect the like terms
Combine $-12x$ and $-12x$ : $-12 + (-12) = -24$ . This is the $2ab$ part of the pattern $(a - b)^2 = a^2 - 2ab + b^2$ , with $a = 3x$ and $b = 4$ .
$9x^2 - 24x + 16$

Answer:

(3x - 4)^2 = 9x^2 - 24x + 16

Check: Substitute

x = 2

. Left side:

(3(2) - 4)^2 = (6 - 4)^2 = (2)^2 = 4

. Right side:

9(2)^2 - 24(2) + 16 = 9(4) - 48 + 16 = 36 - 48 + 16 = 4

. Both sides match, confirming the expansion is correct.

Common mistakes and how to avoid them

Writing

(x + 4)^2 = x^2 + 16

by squaring each term separately and forgetting the middle term entirely.

Correction: Always rewrite the square as

(x + 4)(x + 4)

and apply FOIL. The outer and inner products combine to give the essential middle term

8x

, making the correct answer

x^2 + 8x + 16

.

Losing the negative sign when multiplying, for example writing

(x - 3)(x + 2) = x^2 + 2x - 3x + 6

and then getting

x^2 - x + 6

instead of

x^2 - x - 6

.

Correction: The Last multiplication is

(-3)(+2) = -6

, not

+6

. Treat each subtraction as adding a negative number and carry that sign into every multiplication involving that term.

Only doing two multiplications instead of four, for example computing only First and Last and missing the two middle terms.

Correction: Use FOIL as a four-item checklist. Write down all four partial products separately before you collect any like terms.

Adding exponents when adding like terms, writing

x^2 + 3x^2 = x^4

instead of

4x^2

.

Correction: Exponents are only added when multiplying powers with the same base. When collecting like terms you add the coefficients and keep the exponent the same:

x^2 + 3x^2 = 4x^2

.

Forgetting that the middle term of

(a - b)^2

is negative, and writing

(x - 5)^2 = x^2 + 10x + 25

.

Correction: In

(x - 5)(x - 5)

, the outer product is

-5x

and the inner product is also

-5x

. Together they give

-10x

, making the correct expansion

x^2 - 10x + 25

.

Lesson summary

To expand a product of two binomials, multiply every term in the first bracket by every term in the second bracket — four multiplications in total.
FOIL (First, Outer, Inner, Last) is a checklist that keeps all four multiplications organised and prevents missed terms.
After multiplying, the outer and inner products are like terms; add their coefficients to get the middle term of the trinomial.
A squared binomial such as $(a + b)^2$ must be rewritten as $(a + b)(a + b)$ before expanding; it is never equal to $a^2 + b^2$ .
The pattern $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ come directly from applying FOIL to identical brackets.
Substitute a test value into both the original and expanded forms to verify your answer quickly and catch sign errors.

Check your understanding

Question 1

What is the expanded and simplified form of

(x + 6)(x - 2)

?

$x^2 + 4x - 12$
$x^2 - 4x - 12$
$x^2 + 4x + 12$
$x^2 + 8x - 12$

Show answer and explanation

x^2 + 4x - 12

FOIL gives

x^2 - 2x + 6x - 12

. The like terms

-2x

and

+6x

combine to

+4x

, giving

x^2 + 4x - 12

. Option B has the wrong sign on the middle term. Option C has the wrong sign on the constant. Option D uses the sum

6 + 2

instead of the correct outer-plus-inner combination.

Question 2

Which expression is equal to

(x - 7)^2

?

$x^2 + 49$
$x^2 - 49$
$x^2 - 14x + 49$
$x^2 + 14x - 49$

Show answer and explanation

x^2 - 14x + 49

Rewrite as

(x - 7)(x - 7)

. FOIL:

x^2 - 7x - 7x + 49 = x^2 - 14x + 49

. Option A forgets the middle term entirely and makes the constant positive without subtracting. Option B is the difference of squares pattern, which does not apply here. Option D has the wrong signs on both the middle and last terms.

Question 3

Expand

(3x + 2)(2x - 1)

and choose the correct trinomial.

$6x^2 + x - 2$
$6x^2 - x - 2$
$6x^2 + x + 2$
$5x^2 + x - 2$

Show answer and explanation

6x^2 + x - 2

First:

3x \cdot 2x = 6x^2

. Outer:

3x \cdot (-1) = -3x

. Inner:

2 \cdot 2x = 4x

. Last:

2 \cdot (-1) = -2

. Middle term:

-3x + 4x = +x

. Result:

6x^2 + x - 2

. Option B uses

-x

as the middle term, reversing the sign. Option C has the wrong sign on the constant. Option D incorrectly adds the leading coefficients instead of multiplying them.

Question 4

A student claims

(2x + 5)^2 = 4x^2 + 25

. What is missing?

The $x^2$ term should be $2x^2$ .
The middle term $20x$ has been left out.
The constant should be $-25$ because of the squaring.
Nothing is missing; the student is correct.

Show answer and explanation

The middle term

20x

has been left out.

Expanding

(2x + 5)(2x + 5)

with FOIL gives

4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25

. The student squared each term individually and skipped the outer and inner products, losing the middle term

20x

. The constant

25

is positive, and the

x^2

coefficient of

4

is correct.

Key terms

Binomial: An algebraic expression that contains exactly two terms joined by addition or subtraction, such as $3x + 4$ or $x - 7$ .
Trinomial: An algebraic expression that contains exactly three terms, such as $x^2 + 5x + 6$ . Expanding two binomials usually produces a trinomial.
Expand: To remove brackets by multiplying, writing a product of brackets as a sum of individual terms.
FOIL: A checklist for multiplying two binomials: First, Outer, Inner, Last — the four pairs of terms that must be multiplied together.
Like terms: Terms that contain exactly the same variable(s) raised to exactly the same power(s). Only like terms can be added or subtracted.
Coefficient: The number multiplied by the variable part of a term. In $-7x$ , the coefficient is $-7$ .
Distributive property: The rule that $a(b + c) = ab + ac$ . Every term inside the bracket is multiplied by the factor outside it.
Partial product: One of the individual multiplication results produced during expansion, before like terms are combined.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic Q2. It is a study resource, not an official curriculum publication.

Official curriculum reference

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Q2 · Expand products and squares of binomials

Ontario Grade 10 Mathematics

Building Trinomials from Two Brackets

What you will learn

Grade 9 Bridge: The Distributive Property

Expanding a Product of Two Binomials

Area Model: Seeing Why It Works

Squaring a Binomial: A Special Case

Putting It All Together: Strategy and Checks

Area Model for (x + 3)(x + 5)

FOIL Checklist at a Glance

Expanding a Product of Two Different Binomials

Expanding a Squared Binomial

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MFM2P

About this lesson