$\begin{array}{l}\phantom{\phantom{;}x^{2}+2;}{-5x\phantom{;}+3\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+2\overline{\smash{)}-5x^{3}+3x^{2}-x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+2;}\underline{\phantom{;}5x^{3}\phantom{-;x^n}+10x\phantom{;}\phantom{-;x^n}}\\\phantom{\phantom{;}5x^{3}+10x\phantom{;};}\phantom{;}3x^{2}+9x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2-;x^n;}\underline{-3x^{2}\phantom{-;x^n}-6\phantom{;}\phantom{;}}\\\phantom{;-3x^{2}-6\phantom{;}\phantom{;}-;x^n;}\phantom{;}9x\phantom{;}-5\phantom{;}\phantom{;}\\\end{array}$

$-5x+3+\frac{9x-5}{x^2+2}$