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

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