$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}6x^{3}+5x^{2}-6x\phantom{;}-4\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}6x^{5}+5x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-6x^{5}\phantom{-;x^n}-6x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-6x^{5}-6x^{3};}\phantom{;}5x^{4}-6x^{3}+x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{-5x^{4}\phantom{-;x^n}-5x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-5x^{4}-5x^{2}-;x^n;}-6x^{3}-4x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n;}\underline{\phantom{;}6x^{3}\phantom{-;x^n}+6x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}6x^{3}+6x\phantom{;}-;x^n-;x^n;}-4x^{2}+6x\phantom{;}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n-;x^n;}\underline{\phantom{;}4x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}4x^{2}+4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}6x\phantom{;}+2\phantom{;}\phantom{;}\\\end{array}$

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