GCSE

Expanding & Factorising

This might look intimidating at first glance, but the method is actually very straightforward once you see the pattern.

What Your Child Is Learning

Your child is learning two skills that are essentially opposites of each other:

Expanding means multiplying out brackets. You take something like $(x+3)(x+2)$ and turn it into a single expression without brackets.
Factorising is the reverse — taking an expression like $x^2 + 5x + 6$ and putting it back into brackets: $(x+3)(x+2)$.

Think of it like wrapping and unwrapping a present. Expanding unwraps the brackets; factorising wraps them back up. The algebra might look unfamiliar, but it is really just organised multiplication.

$$(x+3)(x+2) = x^2 + 5x + 6$$

How Schools Teach It Now

If you did maths at school, you might remember just being told to "multiply everything out." Today, schools use more structured methods to keep things clear:

For expanding, the two most common methods are:

FOIL — stands for First, Outside, Inside, Last. It tells your child which pairs to multiply: $(x+3)(x+2)$$ $$\text{First: } x \times x = x^2$$ $$\text{Outside: } x \times 2 = 2x$$ $$\text{Inside: } 3 \times x = 3x$$ $$\text{Last: } 3 \times 2 = 6$$ $$= x^2 + 2x + 3x + 6 = x^2 + 5x + 6$
Grid method — a simple table that lays out all four multiplications visually. This is especially helpful for students who like to see things laid out neatly.

For factorising, the standard method is:

Look at the expression, for example $x^2 + 5x + 6$.
Find two numbers that multiply to give the last number (6) and add to give the middle number (5).
Those two numbers go in the brackets: $(x+3)(x+2)$.

Parent tip

You do not need to understand all the algebra to help. The key thing to reinforce is: "find two numbers that multiply to give ___ and add to give ___." That is genuinely the entire method whenever there is nothing in front of the $x^2$.

Common Mistakes to Watch For

These are the errors that trip up almost every student at some point:

Classic mistake

Forgetting the middle term when squaring brackets. Students see $(x+3)^2$ and write $x^2 + 9$. The correct answer is:

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

That $6x$ in the middle comes from $x \times 3$ appearing twice. This is probably the single most common expanding error at GCSE.

Watch out

Sign errors when factorising with negatives. When the expression has negative numbers, students often get the signs wrong in the brackets. For example, $x^2 - x - 6$ factorises to $(x-3)(x+2)$, not $(x+3)(x-2)$. The trick is to check: do those two numbers multiply to give $-6$ AND add to give $-1$?

A good habit to build: after factorising, always expand the answer to check it matches the original. It only takes a few seconds and catches most errors.

Try It Together

Here is one to work through with your child. No pressure — just have a go.

Factorise $x^2 + 7x + 12$.

Step by step:

We need two numbers that multiply to give 12 and add to give 7.
Think about factor pairs of 12: $1 \times 12$, $2 \times 6$, $3 \times 4$.
$3 + 4 = 7$. That is the pair we need.
Write the answer: $(x+3)(x+4)$.

$$x^2 + 7x + 12 = (x+3)(x+4)$$

You can check by expanding: $x \times x = x^2$, $x \times 4 = 4x$, $3 \times x = 3x$, $3 \times 4 = 12$. Add them up: $x^2 + 7x + 12$. It matches.

Conversation starter

Ask your child: "What two numbers multiply to give 20 and add to give 9?" (Answer: 4 and 5.) Once they can find those pairs quickly, factorising becomes almost automatic.

Let Them Practise

The best way to build confidence with expanding and factorising is repetition. Once your child spots the patterns, it starts to feel almost like a puzzle.

We have built a free interactive tool where they can practise factorising quadratics with instant feedback — no login, no cost, just maths:

Quadratic Factoriser

Free interactive practice — instant feedback on every answer.

Practise Now on MaffsGames

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