Expanding brackets: single, double and triple

TutorAura guide · Updated 25 October 2026

Expanding brackets means multiplying everything inside a bracket by what's outside it, so the expression has no brackets left. For a single bracket, multiply each term inside by the term outside: $3(x + 4) = 3x + 12$ . For two brackets, multiply every term in the first by every term in the second, then collect like terms.

Key takeaways

Multiply every term inside by the term outside.
Double brackets give four products: use a grid or FOIL.
(x + a)² means (x + a)(x + a), not x² + a².
Watch the signs: a negative outside changes every sign inside.

Single brackets

$3(x + 4) = 3x + 12$
$-2(x - 5) = -2x + 10$ (a negative times a negative is positive)
$4x(2x - 3) = 8x^2 - 12x$
Expand and simplify: $3(x + 2) + 2(x - 1) = 3x + 6 + 2x - 2 = 5x + 4$

Double brackets

$(x + 3)(x + 5)$ with a grid:

×	$x$	$+5$
$x$	$x^2$	$5x$
$+3$	$3x$	$15$

Add the four cells: $x^2 + 8x + 15$ .

FOIL (First, Outer, Inner, Last) gives the same four products:

$(x - 4)(x + 2) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8$
$(2x + 1)(x - 3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3$

Squared brackets and special cases

$(x + 5)^2 = (x + 5)(x + 5) = x^2 + 10x + 25$ , not $x^2 + 25$ .
$(3x - 2)^2 = 9x^2 - 12x + 4$ .
Difference of two squares: $(x + 3)(x - 3) = x^2 - 9$ ; the middle terms cancel.

Spotting these patterns makes factorising much easier later.

Triple brackets

Expand two brackets first, then multiply the result by the third:

$(x + 1)(x + 2)(x + 3)$ : first $(x + 1)(x + 2) = x^2 + 3x + 2$ . Then

(x^2 + 3x + 2)(x + 3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6 = x^3 + 6x^2 + 11x + 6

Triple brackets are GCSE Higher content.

Common mistakes

Multiplying only the first term inside the bracket: $3(x + 4) \neq 3x + 4$ .
Squaring each term: $(x + 5)^2 \neq x^2 + 25$ .
Sign errors with a negative outside: $-2(x - 5) \neq -2x - 10$ .
Forgetting to collect like terms at the end.
Quick check: substitute a number such as $x = 2$ into both forms. They should match.

Practice

$4(2x - 3)$
$(x + 6)(x - 1)$
$(3x - 2)^2$
$(2x + 5)(2x - 5)$

Answers: $8x - 12$ , $x^2 + 5x - 6$ , $9x^2 - 12x + 4$ , $4x^2 - 25$ . For more, see the equivalent expressions topic guide.

Frequently asked questions

What does 'expand and simplify' mean?

Multiply out all the brackets, then collect like terms so each type of term appears once. For example, 2(x + 3) + 4(x − 1) = 2x + 6 + 4x − 4 = 6x + 2.

How do I expand a bracket with a minus sign in front?

Multiply every term inside by the negative number, including its sign. −3(x − 4) = −3x + 12, because a negative times a negative is positive. A bracket with just a minus in front, like −(x − 4), means −1 × (x − 4) = −x + 4.

What does FOIL stand for?

First, Outer, Inner, Last: the four products you need when expanding two brackets that each have two terms. A grid gives the same four products and also works for longer brackets.

Why isn't (x + 5)² equal to x² + 25?

Squaring a bracket means multiplying it by itself, which gives four products: x², 5x, 5x and 25. The two 5x terms add to 10x, so (x + 5)² = x² + 10x + 25.

How do I check my expansion?

Substitute a number for x into the original and the expanded form. With x = 2, (x + 3)(x + 5) = 5 × 7 = 35 and x² + 8x + 15 = 4 + 16 + 15 = 35, so the expansion is right.

Sources

National curriculum in England: mathematics programmes of study. Department for Education.
Common Core State Standards for Mathematics. Common Core State Standards Initiative.
Mathematics GCSE subject content and assessment objectives. Department for Education.

Expanding brackets questions and answers

15 GCSE and IGCSE questions with worked answers, checked as you type.

How to factorise: common factors and quadratics

Common factors, x² + bx + c, the difference of two squares, and quadratics where a isn't 1.

Expanding, factorising and equivalent expressions

Expand and factorise expressions, recognise equivalent forms and simplify algebraic fractions.

Quadratic equation solver with steps

Solve ax² + bx + c = 0 exactly, with the discriminant, factorised form, surds and the vertex.

An AI algebra tutor: from your first equation to quadratics

Equations, graphs, systems, factoring and quadratics, taught step by step with your working checked on a whiteboard.