Question

Solve in positive integers the cubic:
(x^3 + 2^3) - (x + 1)^2 = 2009

Answer 1

Given equation can be simplified to, x(x-2)(x+1)=2002.
Since, the 3 factors are closer to each other, one root may lie closer to $\sqrt[3]{2002}$.
Which means one root is closer to 12.603---. From trial and error, you may find that 13 is one root of equation.