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

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