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

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