What does FOIL stand for?

First, Outer, Inner, Last — the four pairs of terms you multiply when expanding two brackets. It is just a way to make sure you don't miss any pair.

Why does (x + 3)² not equal x² + 9?

Squaring a bracket means multiplying it by itself, so you also get the two middle terms: (x + 3)(x + 3) = x² + 6x + 9. Forgetting the 6x is a very common mistake.

How many terms should I get before simplifying?

Multiplying two two-term brackets always gives four products. Two of them are usually like terms that you then combine, leaving three terms.

Expanding Double Brackets | Math

From single to double brackets

You already know how to expand a single bracket: multiply the term outside by everything inside, as in 3(x + 2) = 3x + 6. (If that is new, see expanding brackets.)

Double brackets look like (x + 2)(x + 3). Now there are two terms on the outside, not one. The rule grows naturally:

Every term in the first bracket must multiply every term in the second bracket.

With two terms in each bracket, that gives 2 × 2 = 4 products.

FOIL: a way to stay organised

FOIL names the four products so you never miss one:

F — First terms (the first term of each bracket)
O — Outer terms (the two on the outside)
I — Inner terms (the two on the inside)
L — Last terms (the last term of each bracket)

Worked example 1: all positive

Expand (x + 2)(x + 3).

F: x × x   = x²
O: x × 3   = 3x
I: 2 × x   = 2x
L: 2 × 3   = 6

Add them up: x² + 3x + 2x + 6. The Outer and Inner terms are like terms, so collect them:

x² + 5x + 6

Worked example 2: a negative term

Expand (x + 4)(x − 1). Keep each sign attached to its term.

F: x × x    = x²
O: x × (−1) = −x
I: 4 × x    = 4x
L: 4 × (−1) = −4

So x² − x + 4x − 4 = x² + 3x − 4.

Worked example 3: two negatives

Expand (x − 5)(x − 2). Remember negative × negative = positive.

F: x × x       = x²
O: x × (−2)    = −2x
I: (−5) × x    = −5x
L: (−5) × (−2) = +10

So x² − 2x − 5x + 10 = x² − 7x + 10.

The grid (box) method — same maths, clearer layout

Some people prefer a 2×2 grid. For (x + 4)(x − 1):

×	x	+4
x	x²	4x
−1	−x	−4

Add the four boxes: x² + 4x − x − 4 = x² + 3x − 4. This is the same as the grid method for multiplication you may have met with numbers — it just uses algebra terms instead of digits.

Worked example 4: difference of two squares

Expand (x + 6)(x − 6).

F: x × x       = x²
O: x × (−6)    = −6x
I: 6 × x       = +6x
L: 6 × (−6)    = −36

The Outer and Inner terms −6x and +6x cancel, leaving:

x² − 36

This neat pattern, (x + a)(x − a) = x² − a², is called the difference of two squares.

Worked example 5: squaring a bracket

Expand (x + 3)².

A square means "times itself", so (x + 3)² = (x + 3)(x + 3):

F: x × x   = x²
O: x × 3   = 3x
I: 3 × x   = 3x
L: 3 × 3   = 9

So x² + 3x + 3x + 9 = x² + 6x + 9.

Common error: (x + 3)² is not x² + 9. You must include the middle term 6x.

Worked example 6: a coefficient in front of x

Expand (2x + 1)(x + 4).

F: 2x × x  = 2x²
O: 2x × 4  = 8x
I: 1 × x   = x
L: 1 × 4   = 4

So 2x² + 8x + x + 4 = 2x² + 9x + 4.

A reliable method

Multiply every term in the first bracket by every term in the second (FOIL or grid).
Keep each + or − sign attached.
You will have four terms; collect the two like (middle) terms.
Write the answer in order: x² term, x term, then the number.

Activity: expand and simplify

(x + 1)(x + 5)
(x + 2)(x − 7)
(x − 3)(x − 4)
(x − 8)(x + 8)
(x + 5)²
(3x + 2)(x + 1)

Answers:

x² + 6x + 5
x² − 5x − 14
x² − 7x + 12
x² − 64 (difference of two squares)
x² + 10x + 25
3x² + 5x + 2

Where this leads

Expanding double brackets is exactly the reverse of factorising a quadratic, and the expressions you produce — like x² + 5x + 6 — are the ones you meet when solving quadratic equations and plotting their graphs.

Expanding Double Brackets

Key takeaways