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

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