Expanding Single & Double Brackets
SPEC A4 • ALGEBRA — EXPANDING BRACKETS
The mechanical reverse of factorising. Master it now — every quadratic, identity, and algebraic proof in the rest of the course depends on it.
What You'll Learn
Expanding means removing brackets by multiplying through. It's the opposite of factorising — and one of the most reliable places to pick up marks on every AQA paper.
This Lesson Covers
Single brackets — the distributive law a(b + c) = ab + ac
Sign tracking — why −2(x − 1) is −2x + 2, not −2x − 2
Combining expansions — "expand and simplify" questions
Double brackets — FOIL and grid methods
Squaring a bracket — and why (x + 4)² is NOT x² + 16
Why it matters
"Expand and simplify" is one of the most reliable 2-mark questions in the AQA paper — the Jun 2023 Paper 1 Q8 asked exactly this with `5(3x + 4) − 2(x − 1)`. It also underpins every factorising, quadratic, and proof question in the rest of Module 2.
Single Brackets: The Distributive Law
The Rule
The term outside the bracket multiplies every term inside the bracket — no matter how many terms there are.a(b + c) = ab + ac
Worked examples
3(2x − 5) = 6x − 15x(x + 3) = x² + 3x4n(n² + 2) = 4n³ + 8n3x(x² + 2x − 5) = 3x³ + 6x² − 15x (three terms inside — still one × each)
Golden rule
One multiplication per term inside. If there are three terms inside, there are three multiplications. Don't skip any.
The Negative Sign Trap
When a negative sits outside the bracket, every sign inside flips. Miss this and AQA will find you.
WEAK STUDENT ✗−2(x − 1) = −2x − 2
Forgot that negative × negative = positive.
STRONG STUDENT ✓−2(x − 1) = −2x + 2
−2 × x = −2x, and −2 × (−1) = +2.
More examples
−2(x − 4) = −2x + 8−5(2a + 3) = −10a − 15−(x − 7) = −x + 7 (a minus sign with no number means coefficient −1)
Remember: the minus belongs to the term on its right
In 3x − 2(x + 4), the −2 multiplies both x and +4. So: 3x − 2x − 8 = x − 8. Writing the −2 explicitly in front of each product keeps the signs safe.
01Worked Example — Authentic AQA Question
Expand and simplify fully: 5(3x + 4) − 2(x − 1) [2 marks]JUN 2023 • P1 • Q8
Step 1 — Expand the first bracket
5(3x + 4) = 15x + 20
Step 2 — Expand the second bracket (watch the signs!)
−2(x − 1) = −2x + 2
Key check: −2 × −1 = +2 , not −2.
Step 3 — Combine and collect like terms
15x + 20 − 2x + 2 = (15x − 2x) + (20 + 2)
= 13x + 22
Real AQA mark scheme (Jun 2023)
B2 for the fully correct answer 13x + 22. B1 for partial progress: 15x + 20, or −2x + 2, or "13x + any number", or "any number × x + 22". The common wrong answer 13x + 18 came from getting the second bracket's sign wrong.
Double Brackets: The FOIL Method
With two brackets, each term in the first multiplies each term in the second — four products in total. Use the acronym FOIL to make sure you get all four.
(x + 3)(x + 5)
First: x · x = x²Outer: x · 5 = 5xInner: 3 · x = 3xLast: 3 · 5 = 15
Step 2 — collect like terms
x² + 5x + 3x + 15 = x² + 8x + 15
Why FOIL always works
FOIL is just the distributive law applied twice — once per term in the first bracket. That's why it guarantees exactly four products for two binomials.
Double Brackets: The Grid Method
Same answer, more visual. Write the terms of each bracket on the edges of a 2×2 grid. Each cell is one product. Then sum the cells.
Example: (x + 3)(x + 5)
×x+5xx²+5x+3+3x+15Sum: x² + 5x + 3x + 15 = x² + 8x + 15
Grid vs FOIL
Both give the same answer. Grid is safer with negatives — the signs live in the boxes and are harder to lose. FOIL is faster once you're confident. Pick one and stick with it in an exam.
Double Bracket with a Negative
The grid method earns its keep when signs get messy. Every cell keeps its sign.
02Worked Example
Expand and simplify: (x + 3)(x − 7) [2 marks]
×x−7
xx²−7x
+3+3x−21
Sum: x² − 7x + 3x − 21
= x² − 4x − 21
Sign check
The "−7" belongs in the column header and goes into every cell in that column. If you write just "7" in the header, the whole column's signs go wrong.
MCE Worked Example
Expand and simplify:
(2x − 3)(x + 4)
[2 marks]
Method
Use FOIL: multiply each term of the first bracket by each term of the second. Track the negative on the 3.
Calculate
F: 2x·x = 2x²
O: 2x·4 = 8x
I: −3·x = −3x
L: −3·4 = −12
2x² + 8x − 3x − 12
Evaluate
= 2x² + 5x − 12
Mark scheme pattern
Typical AQA split for "expand and simplify" a double bracket: B1 for three of the four terms correct (or four terms but one sign wrong); B1 for the fully correct simplified expression. Show all four products before you collect — the method mark is in the working.
Squaring a Bracket: The Middle-Term Trap
AQA examiner reports consistently flag this error. Students square each term separately and call it done. Wrong.
WEAK STUDENT ✗(x + 4)² = x² + 16
Squared each term separately. Lost the middle term. Zero or part marks.
STRONG STUDENT ✓(x + 4)² = x² + 8x + 16
Rewrites as (x + 4)(x + 4) and FOILs. Gets all four products and the 8x middle term.
Rule of thumb
When you see a bracket squared, always rewrite as a product of two brackets first: (expression)² = (expression)(expression). Then expand normally. Never try to shortcut.
The Two Square Identities
Once you've expanded a few squared brackets, a pattern appears. Memorise these two identities — they save time on Paper 1 (non-calculator).
Identity 1 — Sum squared
(a + b)² ≡ a² + 2ab + b²Square the first, plus twice the product, square the last.
Identity 2 — Difference squared
(a − b)² ≡ a² − 2ab + b²Same structure, but the middle term is negative. Both endpoints stay positive (since b² is always positive).
Quick check
(x + 4)² = x² + 2(x)(4) + 16 = x² + 8x + 16 ✓
03Worked Example — Squared Bracket
Expand and simplify: (3p − 4)² [2 marks]
Step 1 — Rewrite as a product
(3p − 4)² = (3p − 4)(3p − 4)
Step 2 — FOIL
F: 3p · 3p = 9p² O: 3p · −4 = −12p
I: −4 · 3p = −12p L: −4 · −4 = +16
Step 3 — Collect like terms
9p² − 12p − 12p + 16
= 9p² − 24p + 16
Identity check
Using (a − b)² = a² − 2ab + b² with a = 3p, b = 4: (3p)² − 2(3p)(4) + 4² = 9p² − 24p + 16. ✓ Same answer.
Examiner Alerts
Alert 1 — Squared bracket loses the middle term
Students write: (x + 4)² = x² + 16 — squared each term separately.
Write instead: (x + 4)² = (x + 4)(x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16. Always expand fully — never shortcut.
Alert 2 — Negative sign on expansion (Jun 2023 P1 Q8)
Students write: −2(x − 1) = −2x − 2 — forgot that negative × negative = positive.
Write instead: −2(x − 1) = −2x + 2. Multiply the −2 by every term inside, and track sign changes on each one.
Alert 3 — Further incorrect working drops the mark
Students write: 13x + 22 (correct), then continue to "simplify" to 35x — treating the answer as an equation to solve.
Write instead: Stop at 13x + 22. The Jun 2023 Paper 1 Q8 mark scheme explicitly states: "Do not ignore further working for B2 — e.g. 13x + 22 = 35x scores B1 only". Once you have a simplified expression, stop writing — don't merge unlike terms.
Common Errors
Students often writeCorrect versionWhy it matters
(x + 4)² = x² + 16= x² + 8x + 16Middle term missing — the most-cited AQA error on this topic
−2(x − 1) = −2x − 2= −2x + 2Negative × negative is positive
13x + 22 = 35xStop at 13x + 2213x and 22 are unlike terms — cannot combine
(2x − 3)(x + 4) = 2x² − 12= 2x² + 5x − 12Missed the two x-terms (outer + inner)
Wrote x² + 5x + 3x + 15 and stoppedMust collect like terms → x² + 8x + 15"Expand and simplify " means both steps — no mark otherwise
Practice Question 1
Expand and simplify fully:
4(2x − 1) − 3(x + 5)
[2 marks]
M
Method. Expand each bracket separately (watch the negative on the second), then collect like terms.
C
Calculate.
4(2x − 1) = 8x − 4
−3(x + 5) = −3x − 15
Combine: 8x − 4 − 3x − 15 = (8x − 3x) + (−4 − 15) = 5x − 19
E
Evaluate. Final answer: 5x − 19.
Practice Question 2
Expand and simplify:
(2x − 1)(x + 6)
[2 marks]
M
Method. Use FOIL (or grid method). Four products, then collect the x-terms.
C
Calculate.
F: 2x · x = 2x² O: 2x · 6 = 12x
I: −1 · x = −x L: −1 · 6 = −6
Sum: 2x² + 12x − x − 6
E
Evaluate. Final answer: 2x² + 11x − 6.
Practice Question 3
Expand and simplify:
(p − 6)²
[2 marks]
M
Method. Rewrite as (p − 6)(p − 6) — never shortcut a squared bracket. Then FOIL.
C
Calculate.
F: p · p = p² O: p · −6 = −6p
I: −6 · p = −6p L: −6 · −6 = +36
Sum: p² − 6p − 6p + 36
E
Evaluate. Final answer: p² − 12p + 36.
Lesson Summary
The three types of expansion
TypeMethodExample
Single bracketDistributive law: a(b+c) = ab+ac3(2x−5) = 6x−15
Double bracketFOIL or grid — 4 products, then collect(x+3)(x+5) = x²+8x+15
Squared bracketRewrite as (·)(·) first, then FOIL(x+4)² = x²+8x+16
The two square identities
(a + b)² ≡ a² + 2ab + b²
(a − b)² ≡ a² − 2ab + b²
Top 3 traps to avoid
(x + 4)² ≠ x² + 16 — always expand fully, never square each term separately
−2(x − 1) = −2x + 2 — track signs, negative × negative = positive
Stop at 13x + 22 — unlike terms can't be combined; don't "solve" an expression
MCE reminder
"Expand and simplify" is a two-step instruction: first expand (show every product), then collect like terms. AQA awards B1 for partial progress and B2 for the fully simplified expression. Always show all four FOIL products — the method mark is in the working.
Next: What You'll Learn