Expanding

Video lesson4 worked examples

Theory

Expanding Brackets

You must be able to multiply out algebraic expressions by multiplying every term inside the bracket by the term(s) outside.

Double Brackets

To expand two brackets, you multiply each term in the first bracket by each term in the second bracket. This is often summarised using the acronym "FOIL" (First, Outside, Inside, Last).

Candidates must be able to expand expressions ranging from $(ax+b)(cx+d)$ to multiplying a linear bracket by a trinomial bracket, $(ax+b)(cx^2+dx+e)$ .

The Golden Rule: every term inside the bracket must be multiplied by every term outside it — none left behind. When two brackets meet, be systematic (FOIL for two-by-two, or each term of the first across the whole second), then collect like terms at the end.

⚠️ Common Examiner Traps

Sign errors with a negative outside: $-2(x+5)$ is $-2x-10$ , not $-2x+10$ — the minus multiplies both terms.
Squaring a bracket: $(x+3)^2$ means $(x+3)(x+3) = x^2+6x+9$ , not $x^2+9$ .
Missing the middle terms: in a linear-by-trinomial expansion there are six products before collecting — losing one is the usual slip.
Not collecting like terms: an expanded answer left as a string of terms can cost the final mark; always simplify.

Worked examples

Example 1

Single Brackets

Expand and simplify $3(4x - 2) - 2(x + 5)$ .

Step 1: Expand both brackets: $12x - 6 - 2x - 10$ .

Step 2: Collect like terms: $10x - 16$ .

Example 2

Double Brackets

Expand $(2y + 3)(y - 4)$ .

Step 1: Multiply terms (FOIL): $2y^2 - 8y + 3y - 12$ .

Answer: $2y^2 - 5y - 12$ .

Example 3

Linear by Trinomial

Expand $(x + 3)(2x^2 - x + 4)$ .

Step 1: Multiply $x$ by the trinomial, then $+3$ by the trinomial: $2x^3 - x^2 + 4x + 6x^2 - 3x + 12$ .

Step 2: Collect like terms: $2x^3 + 5x^2 + x + 12$ .

Answer: $2x^3 + 5x^2 + x + 12$ .

Example 4

🔗 Bringing it together

Expand and simplify $(x + 4)^2 - (x - 3)(x + 2)$ .

Step 1: Expand each part separately. Remember the square is a double bracket: $(x+4)^2 = x^2 + 8x + 16$ .

Step 2: Expand the second product: $(x-3)(x+2) = x^2 - x - 6$ .

Step 3: Subtract the whole second bracket — every sign inside it flips:

(x^2 + 8x + 16) - (x^2 - x - 6) = x^2 + 8x + 16 - x^2 + x + 6

Step 4: Collect like terms; the $x^2$ terms cancel:

Answer: $9x + 22$ .

Ready to practise?

17 Expanding Brackets questions with answers and video solutions