How do I expand double brackets like (x + 2)(x + 3)?

Multiply every term in the first bracket by every term in the second. Use FOIL (First, Outer, Inner, Last): x·x + x·3 + 2·x + 2·3 = x² + 5x + 6.

Expanding Brackets in KS3 Maths: A Step-by-Step Guide

Q: What does expanding brackets mean in KS3?

Expanding brackets means multiplying out — replacing an expression like 3(x + 4) with 3x + 12 by applying the distributive property to each term inside.

Q: What is the distributive property?

The rule that says a(b + c) = ab + ac. The number outside the brackets multiplies each term inside, separately.

Expanding brackets means multiplying out: 3(x + 4) becomes 3x + 12. The rule is the distributive property: multiply the outside by each inside term, separately. This guide walks single brackets, double brackets, and the four traps that lose marks in Year 7–9 maths.

The core rule

The distributive property says:

a(b + c) = a·b + a·c

In English: whatever is outside the bracket multiplies each term inside, on its own.

Single brackets: worked examples

Example 1: `3(x + 4)`

Outside: 3
Inside: x and 4
Multiply outside by each: 3·x = 3x, 3·4 = 12
Combine: 3x + 12

Example 2: `5(2y - 3)`

Outside: 5
Inside: 2y and -3
Multiply each: 5·2y = 10y, 5·-3 = -15
Combine: 10y - 15

Example 3: `-2(x - 5)` (the negative trap)

Outside: -2
Inside: x and -5
Multiply each: -2·x = -2x, -2·-5 = +10
Combine: -2x + 10

The double negative -2 × -5 = +10 is where most students slip.

Double brackets: FOIL

For (x + 2)(x + 3):

First: x · x = x² Outer: x · 3 = 3x Inner: 2 · x = 2x Last: 2 · 3 = 6

Combine: x² + 3x + 2x + 6 = x² + 5x + 6

Tip: always collect like terms at the end (3x + 2x = 5x). Forgetting this is a routine mark-loss.

The four traps

Trap 1: Distributing only to the first term

❌ 3(x + 4) = 3x + 4 ✅ 3(x + 4) = 3x + 12

Fix: draw arrows from outside to each inside term before you write anything.

Trap 2: Sign-flip on negatives

❌ -2(x - 3) = -2x - 6 ✅ -2(x - 3) = -2x + 6

Fix: when the outside is negative, write each multiplication on its own line first.

Trap 3: Treating `2x` as `2 + x`

❌ 2(3x) = 5x (no — that's 2 + 3x) ✅ 2(3x) = 6x

Fix: when there are no + or - signs inside, it's multiplication: 2 · 3x = 6x.

Trap 4: Forgetting to collect like terms

❌ x² + 3x + 2x + 6 and stopping there ✅ x² + 5x + 6

Fix: the final answer must be in simplest form. Always look for like terms to combine.

Practice problems (try then check)

Expand 4(x + 5) → 4x + 20
Expand 7(2a - 3) → 14a - 21
Expand -3(x + 2) → -3x - 6
Expand -5(2y - 1) → -10y + 5
Expand (x + 1)(x + 4) → x² + 5x + 4
Expand (x - 2)(x + 5) → x² + 3x - 10

When you'll use this

Expanding brackets shows up everywhere from Year 7 onwards:

Year 7: simple linear expressions
Year 8: linear equations, factorising (the reverse of expanding)
Year 9: quadratic expansions, simultaneous equations
GCSE: binomial expansion, quadratic formula derivation

Don't memorise — internalise. The distributive property is one of the three or four maths rules you'll use every year for the rest of school.

How Professor Pi teaches this

In a tutoring session, Pi never says "the answer is 3x + 12". Pi escalates a hint ladder:

"What kind of operation is this?" → you say "distributive property"
"Right — the rule is multiply outside by each inside term."
"What is 3 × x? And 3 × 4?"
(last resort) "Here's 2(x + 5) = 2x + 10. Now your turn."

You do the final step every time. See the 4-level hint ladder.