$\begin{array}{l}\phantom{\phantom{;}x^{2}-2;}{\phantom{;}x^{4}\phantom{-;x^n}+2x^{2}+x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-2\overline{\smash{)}\phantom{;}x^{6}\phantom{-;x^n}\phantom{-;x^n}+x^{3}-2x^{2}\phantom{-;x^n}+3\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2;}\underline{-x^{6}\phantom{-;x^n}+2x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{6}+2x^{4};}\phantom{;}2x^{4}+x^{3}-2x^{2}\phantom{-;x^n}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2-;x^n;}\underline{-2x^{4}\phantom{-;x^n}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-2x^{4}+4x^{2}-;x^n;}\phantom{;}x^{3}+2x^{2}\phantom{-;x^n}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2-;x^n-;x^n;}\underline{-x^{3}\phantom{-;x^n}+2x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-x^{3}+2x\phantom{;}-;x^n-;x^n;}\phantom{;}2x^{2}+2x\phantom{;}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2-;x^n-;x^n-;x^n;}\underline{-2x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}}\\\phantom{;;;-2x^{2}+4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}2x\phantom{;}+7\phantom{;}\phantom{;}\\\end{array}$

$x^{4}+2x^{2}+x+2+\frac{2x+7}{x^2-2}$