$\begin{array}{l}\phantom{\phantom{;}x^{2}+2x\phantom{;}-1;}{\phantom{;}4x^{3}-8x^{2}+18x\phantom{;}-44\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+2x\phantom{;}-1\overline{\smash{)}\phantom{;}4x^{5}\phantom{-;x^n}-2x^{3}\phantom{-;x^n}-x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}-1;}\underline{-4x^{5}-8x^{4}+4x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-4x^{5}-8x^{4}+4x^{3};}-8x^{4}+2x^{3}\phantom{-;x^n}-x\phantom{;}+6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}-1-;x^n;}\underline{\phantom{;}8x^{4}+16x^{3}-8x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}8x^{4}+16x^{3}-8x^{2}-;x^n;}\phantom{;}18x^{3}-8x^{2}-x\phantom{;}+6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}-1-;x^n-;x^n;}\underline{-18x^{3}-36x^{2}+18x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-18x^{3}-36x^{2}+18x\phantom{;}-;x^n-;x^n;}-44x^{2}+17x\phantom{;}+6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}-1-;x^n-;x^n-;x^n;}\underline{\phantom{;}44x^{2}+88x\phantom{;}-44\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}44x^{2}+88x\phantom{;}-44\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}105x\phantom{;}-38\phantom{;}\phantom{;}\\\end{array}$

$4x^{3}-8x^{2}+18x-44+\frac{105x-38}{x^2+2x-1}$