$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}x^{5}+2x^{4}+4x^{3}+8x^{2}+16x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-30x\phantom{;}-5\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-x^{6}+2x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{6}+2x^{5};}\phantom{;}2x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-30x\phantom{;}-5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-2x^{5}+4x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-2x^{5}+4x^{4}-;x^n;}\phantom{;}4x^{4}\phantom{-;x^n}\phantom{-;x^n}-30x\phantom{;}-5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{-4x^{4}+8x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-4x^{4}+8x^{3}-;x^n-;x^n;}\phantom{;}8x^{3}\phantom{-;x^n}-30x\phantom{;}-5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n;}\underline{-8x^{3}+16x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-8x^{3}+16x^{2}-;x^n-;x^n-;x^n;}\phantom{;}16x^{2}-30x\phantom{;}-5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n-;x^n;}\underline{-16x^{2}+32x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;-16x^{2}+32x\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}2x\phantom{;}-5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-2x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;;;;;-2x\phantom{;}+4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n;}-1\phantom{;}\phantom{;}\\\end{array}$

$x^{5}+2x^{4}+4x^{3}+8x^{2}+16x+2+\frac{-1}{x-2}$