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

$4x^{4}+4x^{3}+4x^{2}+4x+6+\frac{5}{x-1}$