How to Expand Double Brackets (FOIL) — KS3 Maths

To expand double brackets, multiply every term in the first bracket by every term in the second, then collect like terms. The FOIL method — First, Outer, Inner, Last — is a reliable order so you never miss a pair. For example, (x + 2)(x + 3) expands to x² + 5x + 6.

What does FOIL stand for?

FOIL is a memory aid for the four multiplications needed when expanding two brackets, each with two terms.

Letter | Means | For (x + 2)(x + 3)
F | First terms | x × x = x²
O | Outer terms | x × 3 = 3x
I | Inner terms | 2 × x = 2x
L | Last terms | 2 × 3 = 6

How do you expand double brackets step by step?

Use FOIL in order, then tidy up:

First: multiply the first term in each bracket.

Outer: multiply the two outermost terms.

Inner: multiply the two innermost terms.

Last: multiply the last term in each bracket.

Collect the like terms (usually the two middle x terms).

For (x + 2)(x + 3): x² + 3x + 2x + 6, then collect 3x + 2x = 5x, giving x² + 5x + 6.

How do you handle negative terms?

Keep each sign with its term and FOIL exactly as before. Expand (x − 4)(x + 5):

First: x × x = x²

Outer: x × 5 = 5x

Inner: −4 × x = −4x

Last: −4 × 5 = −20

Collect: 5x − 4x = x

Answer: x² + x − 20

The Last step, −4 × 5 = −20, is negative; watch those signs.

Worked example: squaring a bracket

(x + 3)² means (x + 3)(x + 3). A common error is to write x² + 9, which is wrong.

First: x² · Outer: 3x · Inner: 3x · Last: 9

Collect: 3x + 3x = 6x

Answer: x² + 6x + 9

Never just square each term — you must FOIL.

Why does this lead into GCSE?

Expanding double brackets produces a quadratic expression (one with an x² term). Recognising the standard form x² + bx + c is the gateway to factorising and solving quadratics at GCSE, so getting fluent now pays off directly.

How can you check your expansion is correct?

A quick way to check a double-bracket expansion is to substitute a simple number, such as x = 1, into both the original brackets and your expanded answer. They should give the same value. For (x + 2)(x + 3), putting x = 1 into the brackets gives (3)(4) = 12; putting x = 1 into x² + 5x + 6 gives 1 + 5 + 6 = 12. The match confirms the expansion. This substitution check is fast, reliable and catches sign slips that are otherwise easy to miss under exam pressure. Get into the habit of doing it whenever you have time at the end of a question — it converts careless errors into easy marks.

Frequently asked questions

How do you expand double brackets?

Multiply every term in the first bracket by every term in the second, using the FOIL order — First, Outer, Inner, Last — then collect the like terms. For example, (x + 2)(x + 3) becomes x² + 5x + 6.

What is the FOIL method?

FOIL stands for First, Outer, Inner, Last. It is the order in which you multiply the terms when expanding two brackets of two terms each, ensuring you carry out all four multiplications and miss none.

Why is (x + 3)² not x² + 9?

Because (x + 3)² means (x + 3)(x + 3), which expands by FOIL to x² + 6x + 9. Squaring each term separately ignores the two "Outer" and "Inner" products that combine to give the 6x middle term.

What do you do with the two middle terms?

The Outer and Inner products usually both contain x, so they are like terms — you collect them into a single term. In (x + 2)(x + 3), the 3x and 2x combine to 5x.

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