Factor: x² + 7x + 12

Factoring a trinomial x² + bx + c: find two numbers that multiply to c and add to b. Here: multiply to 12, add to 7. Try pairs: 1 × 12 = 12, but 1 + 12 = 13 ✗; 2 × 6 = 12, but 2 + 6 = 8 ✗; 3 × 4 = 12, and 3 + 4 = 7 ✓. So x² + 7x + 12 = (x + 3)(x + 4). Verify by FOIL (First, Outer, Inner, Last): (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓. Signs guide factoring: (1) c positive, b positive — both factors positive; (2) c positive, b negative — both factors negative; (3) c negative — factors have opposite signs (the larger absolute value matches the sign of b). Example with negative signs: x² − 5x + 6 = (x − 2)(x − 3); x² + x − 6 = (x + 3)(x − 2). Factoring is the reverse of FOIL. Used to solve quadratic equations: set the factored form = 0 and use zero-product property.