$\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};}\phantom{;}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{-m^{7}+m^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\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;}\phantom{;}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{-m^{5}+m^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\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;}\phantom{;}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{-m^{3}+m^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\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;}\phantom{;}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{-m\phantom{;}+1\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}$

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