Find the integers $n$ such that $n^{5}-2 n^{4}-7 n^{2}-7 n+3=0$.
The equation can be rewritten as $n\left(n^{4}-2 n^{3}-7 n-7\right)=-3$. In particular, $n \mid -3$. Therefore, $n$ has only 4 possible values: $-3, -1, 1, 3$. We test these values manually, and we find that only -1 and 3 are solutions.