$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}x^{2}-2x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{3}-4x^{2}+6x\phantom{;}-24\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-x^{3}+2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{3}+2x^{2};}-2x^{2}+6x\phantom{;}-24\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{\phantom{;}2x^{2}-4x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}2x^{2}-4x\phantom{;}-;x^n;}\phantom{;}2x\phantom{;}-24\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{-2x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;;-2x\phantom{;}+4\phantom{;}\phantom{;}-;x^n-;x^n;}-20\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-2x+2+\frac{-20}{x-2}$