$\begin{array}{l}\phantom{\phantom{;}x^{2}-2x\phantom{;}+2;}{\phantom{;}x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-2x\phantom{;}+2\overline{\smash{)}\phantom{;}x^{3}\phantom{-;x^n}+6x\phantom{;}+3\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+2;}\underline{-x^{3}+2x^{2}-2x\phantom{;}\phantom{-;x^n}}\\\phantom{-x^{3}+2x^{2}-2x\phantom{;};}\phantom{;}2x^{2}+4x\phantom{;}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+2-;x^n;}\underline{-2x^{2}+4x\phantom{;}-4\phantom{;}\phantom{;}}\\\phantom{;-2x^{2}+4x\phantom{;}-4\phantom{;}\phantom{;}-;x^n;}\phantom{;}8x\phantom{;}-1\phantom{;}\phantom{;}\\\end{array}$

$x+2+\frac{8x-1}{x^2-2x+2}$