$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}x^{2}-3x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{3}-5x^{2}-2x\phantom{;}+24\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-x^{3}+2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{3}+2x^{2};}-3x^{2}-2x\phantom{;}+24\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{\phantom{;}3x^{2}-6x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}3x^{2}-6x\phantom{;}-;x^n;}-8x\phantom{;}+24\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{\phantom{;}8x\phantom{;}-16\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}8x\phantom{;}-16\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}8\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-3x-8+\frac{8}{x-2}$