When the coefficient of $x^2$ is greater than 1 (i.e. $a > 1$), factorising requires more care.
Method — find two numbers that multiply to $ac$ and add to $b$, then split the middle term:
Example: $6x^2 + 11x + 4$
- $a = 6$, $b = 11$, $c = 4$, so $ac = 24$
- Find two numbers that multiply to 24 and add to 11: 3 and 8
- Split: $6x^2 + 3x + 8x + 4$
- Factorise in pairs: $3x(2x + 1) + 4(2x + 1)$
- Result: $(3x + 4)(2x + 1)$
Check by expanding: $(3x + 4)(2x + 1) = 6x^2 + 3x + 8x + 4 = 6x^2 + 11x + 4$ ✓
Alternative — trial and improvement: write $(px + q)(rx + s)$ and test factor pairs of $a$ and $c$.
If factorising is not straightforward, use the quadratic formula instead.
Common error: only looking for factors of $c$ rather than factors of $ac$.