Expanding Binomials
Two brackets, four products, one answer. Learn FOIL, the area model, and the perfect-square shortcut — then apply them to any binomial product.
Printable Worksheets
Print or save as PDF — or build a custom worksheet from any module's questions.
Before you read on — expand $(x + 3)(x + 2)$. How many separate multiplications did you do? Try it, then check your reasoning as you go.
When two binomials sit side by side, every term in the first bracket must multiply every term in the second. That gives exactly four partial products. Add them, collect like terms, and you're done.
An expansion of $(a+b)(c+d)$ produces four terms: $ac$, $ad$, $bc$, and $bd$. The middle two usually combine. FOIL is just a memory hook for the four pairs.
Know
- The FOIL method and what each letter stands for
- The area-model interpretation of $(a+b)(c+d)$
- The perfect-square pattern $(a+b)^2 = a^2 + 2ab + b^2$
Understand
- Why there are exactly four partial products
- Why perfect squares produce three terms, not two
- How negative signs affect each partial product
Can Do
- Expand any binomial product using FOIL
- Recognise and apply the perfect-square shortcut
- Handle negative terms without sign errors
Wrong: "$(x + 2)^2 = x^2 + 4$" — squared each term separately.
Right: $(x + 2)^2 = x^2 + 4x + 4$. The middle term $2 \times x \times 2 = 4x$ comes from Outer + Inner.
Wrong: "$(x - 3)(x + 5) = x^2 + 2x + 15$" — sign error on the Last term.
Right: $(x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15$. The $-3 \times +5$ gives $-15$.
FOIL is just a way to remember the four multiplications: First, Outer, Inner, Last. Write all four down, then collect like terms. No shortcuts.
Label the terms in $(a+b)(c+d)$: First is $a \times c$, Outer is $a \times d$, Inner is $b \times c$, Last is $b \times d$. Add them: $ac + ad + bc + bd$.
= ac + ad + bc + bd
Example in action: $(x + 3)(x + 2)$
$$\text{First: } x^2 \quad \text{Outer: } 2x \quad \text{Inner: } 3x \quad \text{Last: } 6$$
$$x^2 + 2x + 3x + 6 = x^2 + 5x + 6$$
Picture $(a+b)(c+d)$ as a big rectangle split into four smaller rectangles. Each small rectangle is one partial product. The total area is the expansion — no algebra needed to see why it works.
The height splits into $a$ and $b$. The width splits into $c$ and $d$. Multiply height-piece by width-piece for each of the four cells. Add the four cell areas.
When both brackets are identical — $(a+b)^2$ — the Outer and Inner products are the same. They combine into one middle term: $2ab$. This gives the famous three-term pattern.
$(a+b)^2 = a^2 + 2ab + b^2$. The $a^2$ is the big square, the $2ab$ is the two identical rectangles, and the $b^2$ is the small corner square. Never drop the middle term.
= a² + 2ab + b²
Example: $(x + 4)^2 = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$.
Using the shortcut: $a = x$, $b = 4$, so $a^2 = x^2$, $2ab = 8x$, $b^2 = 16$.
Negative terms inside brackets are the #1 source of errors. A minus sign travels with its term into every product it touches. Negative × negative = positive — but you have to actually write both negatives first.
For $(x - 3)(x + 5)$: the $-3$ multiplies the $x$ (giving $-3x$) and the $-3$ multiplies the $+5$ (giving $-15$). Both products keep the minus. Only the $x \times +5$ is positive.
= x² + 2x - 15
Watch Me Solve It · 3 examples
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1First × First$x \times x = x^2$Multiply the first term of each bracket.
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2Outer + Inner$x \times 2 = 2x$ and $3 \times x = 3x$These are the two middle products. They will combine.
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3Last × Last$3 \times 2 = 6$Multiply the second term of each bracket.
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4Collect like terms$x^2 + 2x + 3x + 6 = x^2 + 5x + 6$$2x + 3x = 5x$. The $x^2$ and $6$ have no partners.
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1First × First$x \times x = x^2$
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2Outer + Inner$x \times 5 = 5x$ and $-4 \times x = -4x$The $-4$ keeps its minus sign.
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3Last × Last$-4 \times 5 = -20$Negative × positive = negative. This is the most common error spot.
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4Collect like terms$x^2 + 5x - 4x - 20 = x^2 + x - 20$$5x - 4x = x$. Check: four products became three terms.
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1Rewrite as two brackets$(2x - 3)^2 = (2x - 3)(2x - 3)$A square means the expression multiplied by itself.
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2First × First$2x \times 2x = 4x^2$
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3Outer + Inner$2x \times (-3) = -6x$ and $-3 \times 2x = -6x$Both are negative. Together they make $-12x$.
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4Last × Last + collect$-3 \times (-3) = +9$$4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9$Three terms, as every perfect square must.
FOIL
- First × First
- Outer × Outer
- Inner × Inner
- Last × Last
- Then collect like terms
Area Model
- $(a+b)(c+d)$ = big rectangle
- Split into 4 smaller rectangles
- Each cell = one partial product
- Add all four areas
Perfect Squares
- $(a+b)^2 = a^2 + 2ab + b^2$
- $(a-b)^2 = a^2 - 2ab + b^2$
- Always three terms
- Middle term = 2ab
Sign Rules
- Neg × pos = neg
- Neg × neg = pos
- Carry the sign into every product
- Double-check the Last term
How are you completing this lesson?
Brain Trainer · 4 problems
Four problems mixing FOIL, negatives and perfect squares. Work each one, then reveal the answer to check.
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1 Expand $(x + 5)(x + 2)$.
First: $x^2$, Outer: $2x$, Inner: $5x$, Last: $10$$= x^2 + 7x + 10$ -
2 Expand $(x - 3)(x + 4)$.
First: $x^2$, Outer: $4x$, Inner: $-3x$, Last: $-12$$= x^2 + x - 12$ -
3 Expand $(2x + 1)(x - 2)$.
First: $2x^2$, Outer: $-4x$, Inner: $x$, Last: $-2$$= 2x^2 - 3x - 2$ -
4 Expand and simplify $(x + 2)^2 - (x + 1)(x + 3)$.
$(x+2)^2 = x^2 + 4x + 4$. $(x+1)(x+3) = x^2 + 4x + 3$. Subtract: $(x^2 + 4x + 4) - (x^2 + 4x + 3)$$= 1$
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Expand and simplify $(x + 6)(x - 2)$.
Q7. A rectangular sports court has length $(2x + 5)$ metres and width $(x + 3)$ metres. Use the area model to write and simplify an expression for the area of the court.
Q8. When $(x + 5)(x + k)$ is expanded, the result is $x^2 + 8x + 15$.
(a) Find the value of $k$. (1 mark)
(b) Show your working to verify your answer. (2 marks)
Quick Check
1. B — $x \times x = x^2$, $x \times 3 = 3x$, $4 \times x = 4x$, $4 \times 3 = 12$. Sum: $x^2 + 7x + 12$.
2. B — The student forgot the middle term. $(x+2)^2 = x^2 + 2x + 2x + 4 = x^2 + 4x + 4$.
3. A — $x^2 + 5x - 3x - 15 = x^2 + 2x - 15$.
4. B — $6x^2 + 8x - 3x - 4 = 6x^2 + 5x - 4$.
5. C — $(x-5)^2 = x^2 - 5x - 5x + 25 = x^2 - 10x + 25$.
Show Your Working Model Answers
Q6 (2 marks): $x^2 - 2x + 6x - 12$ [1 for four correct products] $= x^2 + 4x - 12$ [1 for collecting].
Q7 (2 marks): Area $= (2x+5)(x+3) = 2x^2 + 6x + 5x + 15$ [1 for four products] $= 2x^2 + 11x + 15$ m² [1 for simplified].
Q8 (3 marks): (a) $k = 3$ [1]. (b) $(x+5)(x+3) = x^2 + 3x + 5x + 15$ [1] $= x^2 + 8x + 15$, which matches [1].
Working Backwards
Expand and simplify $(2x + 3)(x - 1) - (x - 2)^2$. Show every step, then collect like terms.
Reveal solution
$(2x+3)(x-1) = 2x^2 - 2x + 3x - 3 = 2x^2 + x - 3$.
$(x-2)^2 = x^2 - 4x + 4$.
Subtract: $(2x^2 + x - 3) - (x^2 - 4x + 4) = 2x^2 + x - 3 - x^2 + 4x - 4 = x^2 + 5x - 7$.
FOIL
First, Outer, Inner, Last
Area Model
Four cells = four products
Collect
Combine like terms after expanding
Perfect Square
$(a+b)^2 = a^2 + 2ab + b^2$
Middle Term
Never drop the $2ab$
Signs
Neg × neg = pos
Interactive: Binomial Expander
Visualise double-bracket expansion with an interactive area model. Adjust the terms and watch the four partial products update.