$\begin{array}{l}\phantom{\phantom{;}2m\phantom{;}-1;}{\phantom{;}16m^{4}+8m^{3}+4m^{2}+2m\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;}2m\phantom{;}-1\overline{\smash{)}\phantom{;}32m^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2m\phantom{;}-1;}\underline{-32m^{5}+16m^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-32m^{5}+16m^{4};}\phantom{;}16m^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2m\phantom{;}-1-;x^n;}\underline{-16m^{4}+8m^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-16m^{4}+8m^{3}-;x^n;}\phantom{;}8m^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2m\phantom{;}-1-;x^n-;x^n;}\underline{-8m^{3}+4m^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-8m^{3}+4m^{2}-;x^n-;x^n;}\phantom{;}4m^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2m\phantom{;}-1-;x^n-;x^n-;x^n;}\underline{-4m^{2}+2m\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-4m^{2}+2m\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}2m\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2m\phantom{;}-1-;x^n-;x^n-;x^n-;x^n;}\underline{-2m\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;;;;-2m\phantom{;}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}2\phantom{;}\phantom{;}\\\end{array}$

$16m^{4}+8m^{3}+4m^{2}+2m+1+\frac{2}{2m-1}$