$\begin{array}{l}\phantom{\phantom{;}x^{2}+2;}{-5x\phantom{;}+20\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+2\overline{\smash{)}-5x^{3}+20x^{2}-7x\phantom{;}+3\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+2;}\underline{\phantom{;}5x^{3}\phantom{-;x^n}+10x\phantom{;}\phantom{-;x^n}}\\\phantom{\phantom{;}5x^{3}+10x\phantom{;};}\phantom{;}20x^{2}+3x\phantom{;}+3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2-;x^n;}\underline{-20x^{2}\phantom{-;x^n}-40\phantom{;}\phantom{;}}\\\phantom{;-20x^{2}-40\phantom{;}\phantom{;}-;x^n;}\phantom{;}3x\phantom{;}-37\phantom{;}\phantom{;}\\\end{array}$

$-5x+20+\frac{3x-37}{x^2+2}$