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

$x^{4}+2x^{3}+2x^{2}+2x+\frac{3}{x-1}$