$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}3x^{2}+2x\phantom{;}-4\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}3x^{4}+2x^{3}-x^{2}-x\phantom{;}-16\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-3x^{4}\phantom{-;x^n}-3x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{4}-3x^{2};}\phantom{;}2x^{3}-4x^{2}-x\phantom{;}-16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{-2x^{3}\phantom{-;x^n}-2x\phantom{;}\phantom{-;x^n}}\\\phantom{;-2x^{3}-2x\phantom{;}-;x^n;}-4x^{2}-3x\phantom{;}-16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n;}\underline{\phantom{;}4x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}4x^{2}+4\phantom{;}\phantom{;}-;x^n-;x^n;}-3x\phantom{;}-12\phantom{;}\phantom{;}\\\end{array}$

$3x^{2}+2x-4+\frac{-3x-12}{x^2+1}$