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

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