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

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