$\begin{array}{l}\phantom{\phantom{;}3x^{2}+1;}{\frac{5}{3}x\phantom{;}-\frac{1}{3}\phantom{;}\phantom{;}}\\\phantom{;}3x^{2}+1\overline{\smash{)}\phantom{;}5x^{3}-x^{2}+2x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}3x^{2}+1;}\underline{-5x^{3}\phantom{-;x^n}-\frac{5}{3}x\phantom{;}\phantom{-;x^n}}\\\phantom{-5x^{3}-\frac{5}{3}x\phantom{;};}-x^{2}+\frac{1}{3}x\phantom{;}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}+1-;x^n;}\underline{\phantom{;}x^{2}\phantom{-;x^n}+\frac{1}{3}\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}x^{2}+\frac{1}{3}\phantom{;}\phantom{;}-;x^n;}\frac{1}{3}x\phantom{;}-\frac{5}{3}\phantom{;}\phantom{;}\\\end{array}$

$\frac{5}{3}x-\frac{1}{3}+\frac{\frac{1}{3}x-\frac{5}{3}}{3x^2+1}$