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

$4x^{3}+4x^{2}+3x-1+\frac{-6x+1}{x^2-x+1}$