$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}x^{5}\phantom{-;x^n}-x^{3}-4x^{2}+x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}x^{7}\phantom{-;x^n}\phantom{-;x^n}-4x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-x^{7}\phantom{-;x^n}-x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{7}-x^{5};}-x^{5}-4x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{\phantom{;}x^{5}\phantom{-;x^n}+x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{5}+x^{3}-;x^n;}-4x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n;}\underline{\phantom{;}4x^{4}\phantom{-;x^n}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}4x^{4}+4x^{2}-;x^n-;x^n;}\phantom{;}x^{3}+4x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n-;x^n;}\underline{-x^{3}\phantom{-;x^n}-x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-x^{3}-x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}4x^{2}-x\phantom{;}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n-;x^n-;x^n;}\underline{-4x^{2}\phantom{-;x^n}-4\phantom{;}\phantom{;}}\\\phantom{;;;;-4x^{2}-4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-x\phantom{;}-6\phantom{;}\phantom{;}\\\end{array}$

$x^{5}-x^{3}-4x^{2}+x+4+\frac{-x-6}{x^2+1}$