Factorising by Grouping

When an expression has four terms and no single factor common to all of them, group the terms in pairs, factorise each pair, and look for a shared bracket.

Worked example. Factorise $ax + ay + bx + by$.

• Group: $(ax + ay) + (bx + by)$.
• Factor each pair: $a(x + y) + b(x + y)$.
• Both pairs now share the bracket $(x + y)$, so pull it out: $(a + b)(x + y)$.

The magic is that after factorising each pair, an identical bracket appears in both. That common bracket becomes one factor, and the leftovers become the other.