Factorising by grouping and completing the square

An expression with four terms can often be factorised by grouping. For x² + 4x + bx + 4b, group the first pair and the second pair: x(x + 4) + b(x + 4). Now (x + 4) is common, so the result is (x + 4)(x + b). The two brackets must be the same. If a pair is subtracted, change the signs inside: ax − ay − bx + by = a(x − y) − b(x − y) = (x − y)(a − b). If the first grouping fails, regroup. For x² − 3x − y² + 3y, the pairs (x² − 3x) and (y² − 3y) fail. Group (x² − y²) − (3x − 3y) = (x − y)(x + y) − 3(x − y) = (x − y)(x + y − 3). Always take out a common factor first, and check by expanding.