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

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