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

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