$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-1;}{\phantom{;}x^{4}+x^{3}+4x^{2}+4x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{5}\phantom{-;x^n}+3x^{3}\phantom{-;x^n}\phantom{-;x^n}-6\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-1;}\underline{-x^{5}+x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}+x^{4};}\phantom{;}x^{4}+3x^{3}\phantom{-;x^n}\phantom{-;x^n}-6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n;}\underline{-x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{4}+x^{3}-;x^n;}\phantom{;}4x^{3}\phantom{-;x^n}\phantom{-;x^n}-6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n;}\underline{-4x^{3}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-4x^{3}+4x^{2}-;x^n-;x^n;}\phantom{;}4x^{2}\phantom{-;x^n}-6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n;}\underline{-4x^{2}+4x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-4x^{2}+4x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}4x\phantom{;}-6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n;}\underline{-4x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;;;;-4x\phantom{;}+4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-2\phantom{;}\phantom{;}\\\end{array}$

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