$\begin{array}{l}\phantom{\phantom{;}x^{4}+1;}{\phantom{;}x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;}x^{4}+1\overline{\smash{)}\phantom{;}x^{14}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}x^{4}+1;}\underline{-x^{14}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{14}-x^{10};}-x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}x^{4}+1-;x^n;}\underline{\phantom{;}x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{10}+x^{6}-;x^n;}\phantom{;}x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}x^{4}+1-;x^n-;x^n;}\underline{-x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-x^{6}-x^{2}-;x^n-;x^n;}-x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\end{array}$

$x^{10}-x^{6}+x^{2}+\frac{-x^{2}}{x^4+1}$