$\begin{array}{l}\phantom{\phantom{;}x^{2}+3x\phantom{;}+5;}{\phantom{;}8x^{2}-24x\phantom{;}+32\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3x\phantom{;}+5\overline{\smash{)}\phantom{;}8x^{4}\phantom{-;x^n}\phantom{-;x^n}+15x\phantom{;}-24\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+5;}\underline{-8x^{4}-24x^{3}-40x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-8x^{4}-24x^{3}-40x^{2};}-24x^{3}-40x^{2}+15x\phantom{;}-24\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+5-;x^n;}\underline{\phantom{;}24x^{3}+72x^{2}+120x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}24x^{3}+72x^{2}+120x\phantom{;}-;x^n;}\phantom{;}32x^{2}+135x\phantom{;}-24\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+5-;x^n-;x^n;}\underline{-32x^{2}-96x\phantom{;}-160\phantom{;}\phantom{;}}\\\phantom{;;-32x^{2}-96x\phantom{;}-160\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}39x\phantom{;}-184\phantom{;}\phantom{;}\\\end{array}$

$8x^{2}-24x+32+\frac{39x-184}{x^2+3x+5}$