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

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