$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{-x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}-x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{\phantom{;}x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}x^{4}+x^{2};}\phantom{;}2x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{-2x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{;-2x^{2}-2\phantom{;}\phantom{;}-;x^n;}-4\phantom{;}\phantom{;}\\\end{array}$

$-x^{2}+2+\frac{-4}{x^2+1}$