Factorising: common factor and trinomials

(x + p)(x + q) = x² + (p + q)x + pq. So to factorise x² + bx + c we look for two numbers p and q whose product is c and whose sum is b. For x² + 7x + 12 the pairs with product 12 are 1 × 12, 2 × 6 and 3 × 4, and 3 + 4 = 7, so x² + 7x + 12 = (x + 3)(x + 4). Signs: if c is positive, p and q have the same sign as b (both positive or both negative). If c is negative, p and q have different signs, and the one larger in size takes the sign of b. List the pairs in a table to be sure. If the first coefficient is negative, take out −1 first: −x² + 3x + 40 = −(x² − 3x − 40) = −(x − 8)(x + 5). Always check by expanding.