Factor: x² − 16

Difference of squares pattern: a² − b² = (a + b)(a − b). Here x² − 16 = x² − 4², so a = x and b = 4. Factored: (x + 4)(x − 4). Verify by FOIL: (x + 4)(x − 4) = x² − 4x + 4x − 16 = x² − 16 ✓. The middle terms cancel because of the +/− pattern. Key pattern recognition: a² − b² (difference) factors easily; a² + b² (sum) does NOT factor over real numbers (factors using complex numbers as (a + bi)(a − bi)). Common applications: (1) x² − 25 = (x + 5)(x − 5); (2) 4x² − 9 = (2x + 3)(2x − 3); (3) x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2) — factor repeatedly. Perfect square trinomial patterns to also recognize: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Example: x² + 6x + 9 = (x + 3)².