Find three consecutive integers whose products is 85 larger than the cube of the smallest integer

Let the numbers be x, x + 1 and x + 2.

From the information given:-

x (x + 1) (x + 2) = x^3 + 85

Solving for x:-

x (x^2 + 3x + 2) = x^3 + 85

x^3 + 3x^2 + 2x = x^3 + 85

3x^2 + 2x - 85 = 0

3x^2 - 15x + 17x - 85 = 0

3x (x - 5) + 17 (x - 5) = 0

(3x + 17) (x - 5) = 0

x = - 17/3, 5

We take x = 5 because x is an integer.

Answer: - the 3 integers are 5, 6 and 7.