$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}x^{2}+2x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{3}\phantom{-;x^n}-6x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-x^{3}+2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{3}+2x^{2};}\phantom{;}2x^{2}-6x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-2x^{2}+4x\phantom{;}\phantom{-;x^n}}\\\phantom{;-2x^{2}+4x\phantom{;}-;x^n;}-2x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{\phantom{;}2x\phantom{;}-4\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}2x\phantom{;}-4\phantom{;}\phantom{;}-;x^n-;x^n;}-12\phantom{;}\phantom{;}\\\end{array}$

$x^{2}+2x-2+\frac{-12}{x-2}$