Writing It Up · Hard problem

Look back: the lesson wrote $n^3 + 2n = (n - 1)n(n + 1) + 3n$. The same idea for fifth powers: find $c$ with $n^5 - n = (n - 2)(n - 1)n(n + 1)(n + 2) + c\,(n - 1)n(n + 1)$ for all $n$.