Is there integer :$n>0$ for which $\displaystyle\frac{n^4+(n+1)^4}{n²+(n+1)^2}$ is integer? I have tried to show if 
$$\displaystyle \gcd (n^4+(n+1)^4,n^2+(n+1)^2 )=1$$ 
for every positive integer $n$ using standard theorem in number theory as Bèzout and Gauss theorem but I don't succeed. I'm now interesting to seek for a fixed integer $n>0$  for which the ratio: 
$$\displaystyle\frac{n^4+(n+1)^4}{n^2+(n+1)^2}$$
is integer if it is possible  ?
Thank you for any help.

$$
\frac{n^4+(n+1)^4}{n²+(n+1)^2}
=
n^2  + n + \frac32 - \frac1{2 (2 n^2 + 2 n + 1)}
$$
This reduces the question to when is $\displaystyle\frac12 - \frac1{2 (2 n^2 + 2 n + 1)}$ an integer?
The only solutions are $n=0$ and $n=-1$.
