Expanding opens brackets, factorising rebuilds them
What you will learn
Know the rule, then use it
These are the short notes. Read each one, then check you can use it in the worked example below.
Method
FOIL or grid method: multiply every term in the first bracket by every term in the second
Use the grid method (or FOIL)
Draw a 2×2 grid
Write all four products
2x² − 8x + 3x − 12
Collect like terms
−8x + 3x = −5x
Watch out
The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms)
Take out the highest common factor.
Expand and simplify (2x + 3)(x − 4)
Use the grid method (or FOIL): Draw a 2×2 grid. Multiply each pair: First: 2x × x = 2x². Outer: 2x × (−4) = −8x.Inner: 3 × x = 3x. Last: 3 × (−4) = −12.
Write all four products: 2x² − 8x + 3x − 12.We must get 4 terms before collecting because every term in one bracket multiplies every term in the other.
Collect like terms: −8x + 3x = −5x. Final answer: 2x² − 5x − 12.
2x² − 5x − 12
Build up to the hardest questions
Do them in order. If you miss a step, read the solution, then redo the question without looking.
WorkedreasoningExpand and simplify (2x + 3)(x − 4)
3 marks4 minsexpanding-and-factorising-workedShow solution
Expand and simplify (2x + 3)(x − 4)
- 1.Use the grid method (or FOIL): Draw a 2×2 grid. Multiply each pair: First: 2x × x = 2x². Outer: 2x × (−4) = −8x.Inner: 3 × x = 3x. Last: 3 × (−4) = −12.
- 2.Write all four products: 2x² − 8x + 3x − 12.We must get 4 terms before collecting because every term in one bracket multiplies every term in the other.
- 3.Collect like terms: −8x + 3x = −5x. Final answer: 2x² − 5x − 12.
2x² − 5x − 12
- M1: use the grid method (or foil)
- M1: write all four products
- M1: collect like terms
- A1: 2x² − 5x − 12
The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms).This misses the 'cross terms' that combine to give the middle term.The grid method forces 4 separate multiplications and makes it impossible to miss a term.
DiagnosticrecallExpand 4(3x − 2)
1 mark2 minsexpanding-and-factorising-q1Show solution
Expand 4(3x − 2)
- 1.Spot the skill: FOIL or grid method: multiply every term in the first bracket by every term in the second.
- 2.Use the use the grid method (or foil) stage first, then write all four products.
- 3.Keep the final answer visible: 12x − 8.
12x − 8
- M1: use the correct foil or grid method: multiply every term in the first bracket by every term in the second.for double brackets, there are always 4 multiplications. collect like terms at the end.
- A1: 12x − 8
The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms).This misses the 'cross terms' that combine to give the middle term.The grid method forces 4 separate multiplications and makes it impossible to miss a term.
EasyprocedureExpand (x + 5)(x − 3)
2 marks3 minsexpanding-and-factorising-q2Show solution
Expand (x + 5)(x − 3)
- 1.Spot the skill: FOIL or grid method: multiply every term in the first bracket by every term in the second.
- 2.Use the write all four products stage first, then collect like terms.
- 3.Keep the final answer visible: x² + 2x − 15.
x² + 2x − 15
- M1: use the correct foil or grid method: multiply every term in the first bracket by every term in the second.for double brackets, there are always 4 multiplications. collect like terms at the end.
- A1: x² + 2x − 15
The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms).This misses the 'cross terms' that combine to give the middle term.The grid method forces 4 separate multiplications and makes it impossible to miss a term.
MediumreasoningFactorise x² − 7x + 12
3 marks4 minsexpanding-and-factorising-q3Show solution
Factorise x² − 7x + 12
- 1.Spot the skill: FOIL or grid method: multiply every term in the first bracket by every term in the second.
- 2.Use the collect like terms stage first, then use the grid method (or foil).
- 3.Keep the final answer visible: (x − 3)(x − 4).
(x − 3)(x − 4)
- M1: use the correct foil or grid method: multiply every term in the first bracket by every term in the second.for double brackets, there are always 4 multiplications. collect like terms at the end.
- A1: (x − 3)(x − 4)
The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms).This misses the 'cross terms' that combine to give the middle term.The grid method forces 4 separate multiplications and makes it impossible to miss a term.
Hardproblem solvingFactorise fully 6x² + 9x
3 marks5 minsexpanding-and-factorising-q4Show solution
Factorise fully 6x² + 9x
- 1.Spot the skill: FOIL or grid method: multiply every term in the first bracket by every term in the second.
- 2.Use the use the grid method (or foil) stage first, then write all four products.
- 3.Keep the final answer visible: 3x(2x + 3).
3x(2x + 3)
- M1: use the correct foil or grid method: multiply every term in the first bracket by every term in the second.for double brackets, there are always 4 multiplications. collect like terms at the end.
- A1: 3x(2x + 3)
The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms).This misses the 'cross terms' that combine to give the middle term.The grid method forces 4 separate multiplications and makes it impossible to miss a term.
Exam-stylemulti-stepExpand and simplify (x + 3)² − (x − 1)(x + 2)
4 marks6 minsexpanding-and-factorising-q5Show solution
Expand and simplify (x + 3)² − (x − 1)(x + 2)
- 1.Spot the skill: FOIL or grid method: multiply every term in the first bracket by every term in the second.
- 2.Use the write all four products stage first, then collect like terms.
- 3.Keep the final answer visible: 7x + 7.
7x + 7
- M1: use the correct foil or grid method: multiply every term in the first bracket by every term in the second.for double brackets, there are always 4 multiplications. collect like terms at the end.
- A1: 7x + 7
The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms).This misses the 'cross terms' that combine to give the middle term.The grid method forces 4 separate multiplications and makes it impossible to miss a term.
Grade 9 stretchproblem solvingFactorise 6x2 - 15x fully.
4 marks7 minsfactorise-g9Show solution
Factorise 6x2 - 15x fully.
- 1.Find the highest common factor of both terms.
- 2.Take the common x outside the bracket too.
3x(2x - 5)
- M1: identify 3x
- A1: 3x(2x - 5)
Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.
Switch between skills
Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.
1Expanding and factorising - 2 marksExpand 4(3x − 2)Mark answer
12x − 8
2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer
−4
3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer
3x²y + xy
4Linear equations - 3 marksSolve 3(2x − 1) = 2(x + 5)Mark answer
x =
- I can explain the method for expanding and factorising.
- I can show clear working without skipping key steps.
- I can avoid this mistake: The most common error in double bracket expansion is getting only 2 products instead of 4 (multiplying only the first terms and the last terms).This misses the 'cross terms' that combine to give the middle term.The grid method forces 4 separate multiplications and makes it impossible to miss a term.
This guide follows the AQA GCSE Mathematics 8300 specification. Practice questions are original Learnova questions shaped around official content and exam skills.