$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}5x^{4}+10x^{3}-2x^{2}-4x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}5x^{5}\phantom{-;x^n}-22x^{3}\phantom{-;x^n}\phantom{-;x^n}+20\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-5x^{5}+10x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-5x^{5}+10x^{4};}\phantom{;}10x^{4}-22x^{3}\phantom{-;x^n}\phantom{-;x^n}+20\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-10x^{4}+20x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-10x^{4}+20x^{3}-;x^n;}-2x^{3}\phantom{-;x^n}\phantom{-;x^n}+20\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{\phantom{;}2x^{3}-4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}2x^{3}-4x^{2}-;x^n-;x^n;}-4x^{2}\phantom{-;x^n}+20\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n;}\underline{\phantom{;}4x^{2}-8x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}4x^{2}-8x\phantom{;}-;x^n-;x^n-;x^n;}-8x\phantom{;}+20\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}8x\phantom{;}-16\phantom{;}\phantom{;}}\\\phantom{;;;;\phantom{;}8x\phantom{;}-16\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}4\phantom{;}\phantom{;}\\\end{array}$

$5x^{4}+10x^{3}-2x^{2}-4x-8+\frac{4}{x-2}$