For what integer $n$ is $1+n$ divisible by $1+n^2$? For what integer $n$ is $1+n$ divisible by $1+n^2$?

I tried it in modulo $3$ and $8$ but to no avail. 

Trial and error gave me answers $0$,$1$,$-2$ and $-3$.
HINT:

$$\dfrac{n^2+1}{n+1}=\dfrac{(n+1-1)^2+1}{n+1}=n+1-2+\dfrac2{n+1}$$

OR

$$\dfrac{n^2+1}{n+1}=\dfrac{(n+1)(n-1)+2}{n+1}=n-1+\dfrac2{n+1}$$
The only solutions are 0 and 1