$\begin{array}{l}\phantom{-x\phantom{;}+1;}{-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}-x^{2}-x\phantom{;}-1\phantom{;}\phantom{;}}\\-x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x\phantom{;}+1;}\underline{-x^{8}+x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{8}+x^{7};}\phantom{;}x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{-x\phantom{;}+1-;x^n;}\underline{-x^{7}+x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{7}+x^{6}-;x^n;}\phantom{;}x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{-x\phantom{;}+1-;x^n-;x^n;}\underline{-x^{6}+x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-x^{6}+x^{5}-;x^n-;x^n;}\phantom{;}x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{-x\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{-x^{5}+x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-x^{5}+x^{4}-;x^n-;x^n-;x^n;}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{-x\phantom{;}+1-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;-x^{4}+x^{3}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{-x\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{3}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;-x^{3}+x^{2}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{-x\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{2}+x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;;-x^{2}+x\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x\phantom{;}\phantom{-;x^n}\\\phantom{-x\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;;;;;;;-x\phantom{;}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}1\phantom{;}\phantom{;}\\\end{array}$

$-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}-x^{2}-x-1+\frac{1}{1-x}$