What is the minimum value of ((a² + 3a + 1)/a) × ((b² + 3b + 1)/b) for a, b > 0?

(a² + 3a + 1)/a = a + 3 + 1/a. By AM-GM, a + 1/a ≥ 2, so the expression ≥ 5, with equality when a = 1. Similarly (b² + 3b + 1)/b ≥ 5 with equality at b = 1. Hence the product has minimum value 5 × 5 = 25.