Splitting the middle term

Factoring without tiles. Match x² + 7x + 12 with x² + (a + b)x + ab: we need a + b = 7 and ab = 12. Only 3 and 4 work, so it is (x + 3)(x + 4).

Search the factor pairs of the constant. For x² + 11x + 30:

• 2 × 15: product right, sum 17;
• 3 × 10: product right, sum 13;
• 5 × 6: product 30 and sum 11.

So x² + 11x + 30 = (x + 5)(x + 6).

Read the signs first:

• constant positive, middle positive → both numbers positive;
• constant positive, middle negative → both negative: x² – 5x + 6 = (x – 2)(x – 3);
• constant negative → one positive, one negative, and the bigger one takes the middle term's sign.

Identities make products easy too. 23 × 17 = (20 + 3)(20 – 3) = 400 – 9 = 391. 97² = (100 – 3)² = 9409.

Different routes can both be right. James expands (a – b)² first; Reshma groups (a – b)(a + b) first. Both reach a³ – a²b – ab² + b³.

Always expand to check.