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

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