Practice Maths

Expanding Single and Double Brackets

Key Ideas

Key Terms

expanding
Removing brackets by multiplying each term inside by the factor outside — the reverse of factorising.
distributive law
The rule a(b + c) = ab + ac — the outside factor multiplies every term inside the brackets.
binomial
An algebraic expression with exactly two terms, such as (x + 3) or (2x − 5).
FOIL
A method for expanding two binomials: multiply First, Outer, Inner, and Last term pairs in order.
Type Rule Example
Single bracketa(b + c) = ab + ac3(x + 4) = 3x + 12
Negative outside−a(b + c) = −ab − ac−2(x + 5) = −2x − 10
Double brackets (FOIL)(a+b)(c+d) = ac + ad + bc + bd(x+2)(x+3) = x²+5x+6
Expand & simplifyExpand then collect like terms−5x + 3x = −2x
Hot Tip When the term outside a bracket is negative, every sign inside the bracket flips. Double-check sign changes carefully — this is the most common source of errors.

Worked Example — FOIL

Question: Expand (2x + 3)(x − 4).

F: 2x × x = 2x2   |   O: 2x × (−4) = −8x   |   I: 3 × x = 3x   |   L: 3 × (−4) = −12

Collect like terms: 2x2 − 8x + 3x − 12 = 2x2 − 5x − 12

The Distributive Law: Expanding Single Brackets

The distributive law says: a(b + c) = ab + ac. To expand a single bracket, multiply the term outside by every term inside.

Example: 3(x + 5) = 3x + 15. Example: −2(x − 4) = −2x + 8. Note carefully: when the outside term is negative, all signs inside the bracket change.

Example with variables: 5x(2x − 3) = 10x2 − 15x. Multiply coefficients together and add exponents for the variable.

Expanding Two Brackets: FOIL

To expand (a + b)(c + d), multiply each term in the first bracket by each term in the second. The FOIL method gives you a systematic order: First, Outer, Inner, Last.

(x + 3)(x + 5) = x·x + x·5 + 3·x + 3·5 = x2 + 5x + 3x + 15 = x2 + 8x + 15.

(x − 4)(x + 2) = x2 + 2x − 4x − 8 = x2 − 2x − 8.

The Grid Method as an Alternative

The grid method organises the multiplication into a table and is especially useful when you have more terms or when signs get confusing.

For (2x + 3)(x − 5): draw a 2×2 grid. Top row labels: 2x and +3. Left column labels: x and −5. Fill in: 2x·x = 2x2, 3·x = 3x, 2x·(−5) = −10x, 3·(−5) = −15. Sum: 2x2 + 3x − 10x − 15 = 2x2 − 7x − 15.

Collecting Like Terms After Expanding

After expanding, always collect like terms to simplify. Like terms have exactly the same variable part (including exponent).

x2 and 3x2 are like terms. x2 and x are not. Always write the simplified result in descending order of powers: x2 terms first, then x terms, then constants.

Most common mistake: Forgetting to multiply every term. Students often expand 3(x + 5) correctly as 3x + 15, but then make an error like (x + 2)(x + 3) = x2 + 6, forgetting the middle terms. With double brackets, FOIL always gives four terms before collecting — if you get fewer, you have missed a multiplication. Check by counting: 2 terms × 2 terms = 4 products.

Mastery Practice

  1. Expand each single-bracket expression. Fluency

     ExpressionExpanded Form
    (a)3(x + 5) 
    (b)4(2y − 3) 
    (c)−2(a + 7) 
    (d)5(3m − 4) 
    (e)x(x + 6) 
    (f)2a(a − 3) 
    (g)−3p(2p + 1) 
    (h)4n(n − 5) 
    (i)6(2x + 3y − 1) 
    (j)−5k(3k − 2) 

  2. Expand and simplify each double-bracket expression using FOIL. Fluency

     ExpressionExpanded & Simplified
    (a)(x + 3)(x + 2) 
    (b)(y + 5)(y − 1) 
    (c)(a − 4)(a − 3) 
    (d)(2x + 1)(x + 4) 
    (e)(3m − 2)(m + 5) 
    (f)(x + 7)(x − 7) 
    (g)(2a + 3)(2a − 3) 
    (h)(4p + 1)(p − 2) 
    (i)(3k − 4)(2k + 3) 
    (j)(5x − 2)(x − 6) 

  3. Expand and simplify by collecting all like terms. Fluency

     ExpressionSimplified Answer
    (a)2(x + 3) + 3(x − 1) 
    (b)5(a − 2) − 2(a + 4) 
    (c)(x + 4)(x + 1) + 2x 
    (d)(2m + 1)(m + 3) − m2 
    (e)(x + 5)(x − 2) + (x − 3)(x + 1) 
    (f)3x(x + 2) − (x + 1)(x − 4) 

  4. True or False? Write T or F and correct any false statements. Fluency

     StatementT / F
    (a)3(x + 4) = 3x + 4 
    (b)−2(3x − 5) = −6x + 10 
    (c)(x + 2)2 = x2 + 4 
    (d)(x + 3)(x − 3) = x2 − 9 

  5. Rectangle area using algebra. Understanding

    Building dimensions. A rectangular room has length (2x + 5) cm and width (x + 3) cm.
    1. Write an expression for the area of the room by expanding the brackets.
    2. If x = 4, find the area in cm2.
    3. Write and simplify an expression for the perimeter of the room.

  6. Expand using the special product rules. Understanding

    Spotting patterns. Expand the following using the most efficient method and check each answer using FOIL.
    1. (x + 4)2
    2. (y − 6)2
    3. (2a + 5)2
    4. (x + 9)(x − 9)
    5. (4p + 3)(4p − 3)

  7. Spot the mistake. Understanding

    Common errors. Each piece of working below contains exactly one error. Identify the mistake and write the correct answer.
    1. Jordan writes: 3(x + 4) = 3x + 4.
    2. Priya writes: (x + 2)2 = x2 + 4.
    3. Marcus writes: −2(3x − 5) = −6x − 10.
    4. Anya writes: (x + 5)(x − 5) = x2 + 10x − 25.

  8. Square garden and consecutive numbers. Understanding

    Algebra in action. Two consecutive even integers can be written as 2n and 2n + 2. A square garden has side length (3a − 2) cm.
    1. Expand (3a − 2)2 to find an expression for the garden’s area.
    2. Find the area when a = 4.
    3. Show that the product of two consecutive even integers (2n)(2n + 2) expands to 4n2 + 4n.
    4. Use this result with n = 4 to calculate 8 × 10 without direct multiplication.

  9. Finding the width from the perimeter. Problem Solving

    Unknown dimensions. A rectangle has length (x + 6) cm and a perimeter of (6x + 4) cm.
    1. Using P = 2(l + w), write an equation and solve to find an expression for the width.
    2. Verify your expression for width is correct by substituting x = 5 and checking the perimeter.
    3. Find the area of the rectangle when x = 5.

  10. Multi-step expansion challenge. Problem Solving

    Building complexity. An L-shaped room is made from two rectangles: one measuring (x + 4) by (x + 2) and another measuring (x + 1) by (x − 1).
    1. Expand (x + 4)(x + 2) and simplify.
    2. Expand (x + 1)(x − 1) and simplify.
    3. Write a simplified expression for the total area of the L-shaped room (sum of both rectangles).
    4. If x = 6, find the total area.
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