Explanation

Factorise: x² + 5x + 6 = (x + 2)(x + 3) and x² + 8x + 15 = (x + 3)(x + 5). The highest common factor is (x + 3), so comparing with (x + k) gives k = 3. No larger common factor exists, since x + 2 and x + 5 share nothing with each other's co-factors.