Find all integers $n \in \mathbb{Z}$ such that $n+1 \mid n^{2}+1$.
The integer $n$ satisfies the required property if and only if $n+1 \mid n^{2}+1-(n+1)(n-1)=2$. The divisors of 2 are $-2,-1,1,2$ so the solutions are $-3,-2,0,1$.