$\begin{array}{l}\phantom{\phantom{;}x^{2}-3;}{\phantom{;}x^{4}\phantom{-;x^n}+3x^{2}\phantom{-;x^n}+8\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-3\overline{\smash{)}\phantom{;}x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-x^{2}\phantom{-;x^n}-3\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-3;}\underline{-x^{6}\phantom{-;x^n}+3x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{6}+3x^{4};}\phantom{;}3x^{4}\phantom{-;x^n}-x^{2}\phantom{-;x^n}-3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-3-;x^n;}\underline{-3x^{4}\phantom{-;x^n}+9x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-3x^{4}+9x^{2}-;x^n;}\phantom{;}8x^{2}\phantom{-;x^n}-3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-3-;x^n-;x^n;}\underline{-8x^{2}\phantom{-;x^n}+24\phantom{;}\phantom{;}}\\\phantom{;;-8x^{2}+24\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}21\phantom{;}\phantom{;}\\\end{array}$

$x^{4}+3x^{2}+8+\frac{21}{x^2-3}$