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

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