Fancy Functions

Algebra Level 5

For how many positive integers \(b<1000\) do there exist integers \(a\) and \(c\) and non-constant real polynomials \(f(x)\) and \(g(x)\) such that \((f(x))^3+af(x)+b=(g(x))^3+c(g(x))^2\) for all \(x?\)

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