$\begin{array}{l}\phantom{\phantom{;}x^{2}-4;}{\phantom{;}x^{3}\phantom{-;x^n}+4x\phantom{;}\phantom{-;x^n}}\\\phantom{;}x^{2}-4\overline{\smash{)}\phantom{;}x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}x^{2}-4;}\underline{-x^{5}\phantom{-;x^n}+4x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}+4x^{3};}\phantom{;}4x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}-4-;x^n;}\underline{-4x^{3}\phantom{-;x^n}+16x\phantom{;}\phantom{-;x^n}}\\\phantom{;-4x^{3}+16x\phantom{;}-;x^n;}\phantom{;}16x\phantom{;}\phantom{-;x^n}\\\end{array}$

$x^{3}+4x+\frac{16x}{x^2-4}$