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

$4x^{2}+6x-3+\frac{1}{x^2+1}$