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

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