$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}x^{4}+2x^{3}+2x^{2}+5x\phantom{;}+10\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{5}\phantom{-;x^n}-2x^{3}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-x^{5}+2x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}+2x^{4};}\phantom{;}2x^{4}-2x^{3}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-2x^{4}+4x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-2x^{4}+4x^{3}-;x^n;}\phantom{;}2x^{3}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{-2x^{3}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-2x^{3}+4x^{2}-;x^n-;x^n;}\phantom{;}5x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n;}\underline{-5x^{2}+10x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-5x^{2}+10x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}10x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n-;x^n;}\underline{-10x\phantom{;}+20\phantom{;}\phantom{;}}\\\phantom{;;;;-10x\phantom{;}+20\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}19\phantom{;}\phantom{;}\\\end{array}$

$x^{4}+2x^{3}+2x^{2}+5x+10+\frac{19}{x-2}$