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

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