$\begin{array}{l}\phantom{\phantom{;}x^{2}-2;}{\phantom{;}x^{4}\phantom{-;x^n}+2x^{2}-x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-2\overline{\smash{)}\phantom{;}x^{6}\phantom{-;x^n}\phantom{-;x^n}-x^{3}+2x^{2}\phantom{-;x^n}+3\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2;}\underline{-x^{6}\phantom{-;x^n}+2x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{6}+2x^{4};}\phantom{;}2x^{4}-x^{3}+2x^{2}\phantom{-;x^n}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2-;x^n;}\underline{-2x^{4}\phantom{-;x^n}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-2x^{4}+4x^{2}-;x^n;}-x^{3}+6x^{2}\phantom{-;x^n}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2-;x^n-;x^n;}\underline{\phantom{;}x^{3}\phantom{-;x^n}-2x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}x^{3}-2x\phantom{;}-;x^n-;x^n;}\phantom{;}6x^{2}-2x\phantom{;}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2-;x^n-;x^n-;x^n;}\underline{-6x^{2}\phantom{-;x^n}+12\phantom{;}\phantom{;}}\\\phantom{;;;-6x^{2}+12\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}-2x\phantom{;}+15\phantom{;}\phantom{;}\\\end{array}$

$x^{4}+2x^{2}-x+6+\frac{-2x+15}{x^2-2}$