$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}4x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}4x^{3}-8x^{2}-9x\phantom{;}+7\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-4x^{3}\phantom{-;x^n}-4x\phantom{;}\phantom{-;x^n}}\\\phantom{-4x^{3}-4x\phantom{;};}-8x^{2}-13x\phantom{;}+7\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{\phantom{;}8x^{2}\phantom{-;x^n}+8\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}8x^{2}+8\phantom{;}\phantom{;}-;x^n;}-13x\phantom{;}+15\phantom{;}\phantom{;}\\\end{array}$

$4x-8+\frac{-13x+15}{x^2+1}$