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

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