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

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