Factorising Trinomials

A trinomial $x^2 + bx + c$ factorises into $(x + p)(x + q)$ where

Worked example 1. Factorise $x^2 + 5x + 6$.

Pairs multiplying to 6: $(1,6), (2,3)$. Which pair adds to 5? $2 + 3$. So:

Worked example 2 (negative c). Factorise $x^2 - x - 12$.

Need a product of $-12$ (one factor negative) and a sum of $-1$: the pair is $-4$ and $+3$.

Sign logic: $c$ positive → same signs (both matching $b$'s sign). $c$ negative → opposite signs, and the larger factor takes $b$'s sign.