$\begin{array}{l}\phantom{\phantom{;}3x^{2}+2;}{\frac{4}{3}x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;}3x^{2}+2\overline{\smash{)}\phantom{;}4x^{3}-3x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}3x^{2}+2;}\underline{-4x^{3}\phantom{-;x^n}-\frac{8}{3}x\phantom{;}\phantom{-;x^n}}\\\phantom{-4x^{3}-\frac{8}{3}x\phantom{;};}-3x^{2}-\frac{8}{3}x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}+2-;x^n;}\underline{\phantom{;}3x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}3x^{2}+2\phantom{;}\phantom{;}-;x^n;}-\frac{8}{3}x\phantom{;}+4\phantom{;}\phantom{;}\\\end{array}$

$\frac{4}{3}x-1+\frac{-\frac{8}{3}x+4}{3x^2+2}$