$\begin{array}{l}\phantom{\phantom{;}x^{2}+2;}{\phantom{;}x^{6}\phantom{-;x^n}-2x^{4}\phantom{-;x^n}+2x^{2}\phantom{-;x^n}-11\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+2\overline{\smash{)}\phantom{;}x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-2x^{4}\phantom{-;x^n}-7x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+2;}\underline{-x^{8}\phantom{-;x^n}-2x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{8}-2x^{6};}-2x^{6}\phantom{-;x^n}-2x^{4}\phantom{-;x^n}-7x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2-;x^n;}\underline{\phantom{;}2x^{6}\phantom{-;x^n}+4x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}2x^{6}+4x^{4}-;x^n;}\phantom{;}2x^{4}\phantom{-;x^n}-7x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2-;x^n-;x^n;}\underline{-2x^{4}\phantom{-;x^n}-4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-2x^{4}-4x^{2}-;x^n-;x^n;}-11x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2-;x^n-;x^n-;x^n;}\underline{\phantom{;}11x^{2}\phantom{-;x^n}+22\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}11x^{2}+22\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}27\phantom{;}\phantom{;}\\\end{array}$

$x^{6}-2x^{4}+2x^{2}-11+\frac{27}{x^2+2}$