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

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