$\begin{array}{l}\phantom{\phantom{;}x^{2}-8;}{\phantom{;}3x^{3}\phantom{-;x^n}+24x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-8\overline{\smash{)}\phantom{;}3x^{5}\phantom{-;x^n}\phantom{-;x^n}+4x^{2}+12x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-8;}\underline{-3x^{5}\phantom{-;x^n}+24x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{5}+24x^{3};}\phantom{;}24x^{3}+4x^{2}+12x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-8-;x^n;}\underline{-24x^{3}\phantom{-;x^n}+192x\phantom{;}\phantom{-;x^n}}\\\phantom{;-24x^{3}+192x\phantom{;}-;x^n;}\phantom{;}4x^{2}+204x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-8-;x^n-;x^n;}\underline{-4x^{2}\phantom{-;x^n}+32\phantom{;}\phantom{;}}\\\phantom{;;-4x^{2}+32\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}204x\phantom{;}+24\phantom{;}\phantom{;}\\\end{array}$

$3x^{3}+24x+4+\frac{204x+24}{x^2-8}$