$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}x^{4}\phantom{-;x^n}+x^{2}+2x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{5}-2x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}+4\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-x^{5}+2x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}+2x^{4};}\phantom{;}x^{3}\phantom{-;x^n}\phantom{-;x^n}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-x^{3}+2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{3}+2x^{2}-;x^n;}\phantom{;}2x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{-2x^{2}+4x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-2x^{2}+4x\phantom{;}-;x^n-;x^n;}\phantom{;}4x\phantom{;}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n;}\underline{-4x\phantom{;}+8\phantom{;}\phantom{;}}\\\phantom{;;;-4x\phantom{;}+8\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}12\phantom{;}\phantom{;}\\\end{array}$

$x^{4}+x^{2}+2x+4+\frac{12}{x-2}$