Like terms contain exactly the same letter(s) raised to the same power — $3x$ and $-5x$ are like terms, but $3x$ and $3x^2$ are not, no matter how similar they look. Add or subtract the coefficients of like terms only, leaving everything else untouched:

A useful habit is to underline or circle each family of like terms in a different way before combining them — it stops terms with different letters accidentally getting merged together, especially in longer expressions with three or four different letters mixed in.

Multiply the term outside the bracket by every term inside it:

Multiply every term in the first bracket by every term in the second — four multiplications in total (FOIL: First, Outer, Inner, Last), then collect the two middle (like) terms:

If both brackets contain a minus sign, work through each pair of terms carefully rather than trying to do it in your head — $(x-3)(x-6) = x^2-6x-3x+18 = x^2-9x+18$, where two negatives multiplying together (giving $+18$) is easy to get wrong under time pressure.

Factorising reverses expanding: pull the Highest Common Factor (HCF) of every term — numbers and letters — out to the front, leaving the rest in a bracket.

"Factorise fully" is the key instruction: check the bracket afterwards for anything — a number or a letter — that could still be taken out. This also works when every term is negative, or when the common factor itself is negative — factorising $-4x^2-8x$ as $-4x(x+2)$ is often more useful in later working than leaving a leading negative $x^2$ term.

Factorise fully $12a^2b + 18ab^2$.

Solution

1. The HCF of the numbers $12$ and $18$ is $6$.
2. Every term contains at least one $a$ and one $b$, so the full HCF is $6ab$.
3. Divide each term by $6ab$: $\dfrac{12a^2b}{6ab}=2a$ and $\dfrac{18ab^2}{6ab}=3b$.
4. Write the factorised form:

Check by expanding back: $6ab(2a) = 12a^2b$ and $6ab(3b) = 18ab^2$ ✓ — this quick check catches almost every factorising error before you move on.

Expand and simplify $4(2x-1) - 3(x-5)$.

Solution

1. Expand each bracket separately, being careful with the negative sign in front of the second one:

1. Combine the two results:

1. Collect like terms ($x$-terms together, numbers together):

Expanding both brackets before collecting like terms — rather than trying to subtract the brackets as whole blocks first — keeps each sign change explicit and easy to check.