$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+1;}{\phantom{;}x^{4}-x^{3}-4x^{2}+4x\phantom{;}-4\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{5}\phantom{-;x^n}-5x^{3}\phantom{-;x^n}\phantom{-;x^n}+4\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+1;}\underline{-x^{5}-x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{4};}-x^{4}-5x^{3}\phantom{-;x^n}\phantom{-;x^n}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n;}\underline{\phantom{;}x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{4}+x^{3}-;x^n;}-4x^{3}\phantom{-;x^n}\phantom{-;x^n}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n;}\underline{\phantom{;}4x^{3}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}4x^{3}+4x^{2}-;x^n-;x^n;}\phantom{;}4x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{-4x^{2}-4x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-4x^{2}-4x\phantom{;}-;x^n-;x^n-;x^n;}-4x\phantom{;}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}4x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;;;;\phantom{;}4x\phantom{;}+4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}8\phantom{;}\phantom{;}\\\end{array}$

$x^{4}-x^{3}-4x^{2}+4x-4+\frac{8}{x+1}$