Expanding and factorising are opposites — two ways of writing the same expression. Expanding removes brackets by multiplying out; factorising puts brackets back in. Being able to switch between the two lets you write an expression in whichever form is most useful, a bit like unfolding and refolding a piece of paper.
Why this matters: brackets are how algebra packages things up. You expand to simplify and combine terms, and you factorise to solve quadratics, cancel fractions and spot structure. Almost every harder algebra topic — quadratic equations, algebraic fractions, completing the square — leans on being fluent at both directions here.
Area model: 3(x + 2) = 3x + 6
Expanding a single bracket
Multiply everything inside the bracket by the term outside. Think of the outside number as being shared out to every term within.
3(x + 2): 3 × x = 3x and 3 × 2 = 6, so 3(x + 2) = 3x + 6.
4(2x − 3): 4 × 2x = 8x and 4 × −3 = −12, so 4(2x − 3) = 8x − 12. Notice the sign travels with the term.
The 4 reaches every term — including the −3.
Common mistake: don't forget the second term. 3(x + 2) is not 3x + 2 — the 3 multiplies the 2 as well. Every term inside gets multiplied, not just the first.
Expanding double brackets — FOIL
With two brackets, every term in the first must meet every term in the second. FOIL reminds you to catch all four pairings: First, Outer, Inner, Last. It's like a handshake where everyone in one group greets everyone in the other.
Factorising is expanding in reverse. Find the highest common factor of all the terms and take it outside a bracket, leaving what's left inside.
6x + 9: 3 divides into both → 3(2x + 3). Check by expanding: 3 × 2x = 6x, 3 × 3 = 9. ✓
Factorising: pull the shared factor 3 outside the bracket.
6x² − 9x: both share 3 and an x, so take out 3x → 3x(2x − 3). Always pull out the biggest factor, letters included.
Expand →
← Factorise
3(x + 2)
3x + 6
4(2x + 5)
8x + 20
3x(2x − 3)
6x² − 9x
(x + 3)(x + 5)
x² + 8x + 15
In a nutshell: expand by multiplying out (every term by every term); factorise by taking the common factor out. They undo each other — so you can always check a factorised answer by expanding it back.