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

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