$\begin{array}{l}\phantom{\phantom{;}3x^{2}-2x\phantom{;}-1;}{\phantom{;}x^{2}-2x\phantom{;}+\frac{5}{3}\phantom{;}\phantom{;}}\\\phantom{;}3x^{2}-2x\phantom{;}-1\overline{\smash{)}\phantom{;}3x^{4}-8x^{3}+8x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}3x^{2}-2x\phantom{;}-1;}\underline{-3x^{4}+2x^{3}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{4}+2x^{3}+x^{2};}-6x^{3}+9x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-2x\phantom{;}-1-;x^n;}\underline{\phantom{;}6x^{3}-4x^{2}-2x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}6x^{3}-4x^{2}-2x\phantom{;}-;x^n;}\phantom{;}5x^{2}-2x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-2x\phantom{;}-1-;x^n-;x^n;}\underline{-5x^{2}+\frac{10}{3}x\phantom{;}+\frac{5}{3}\phantom{;}\phantom{;}}\\\phantom{;;-5x^{2}+\frac{10}{3}x\phantom{;}+\frac{5}{3}\phantom{;}\phantom{;}-;x^n-;x^n;}\frac{4}{3}x\phantom{;}+\frac{8}{3}\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-2x+\frac{5}{3}+\frac{\frac{4}{3}x+\frac{8}{3}}{3x^2-2x-1}$