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

$x+1+\frac{5x-5}{x^2-x+1}$