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

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