$\begin{array}{l}\phantom{\phantom{;}x^{2}-2x\phantom{;}+3;}{\phantom{;}2x^{2}+x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-2x\phantom{;}+3\overline{\smash{)}\phantom{;}2x^{4}-3x^{3}-4x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+3;}\underline{-2x^{4}+4x^{3}-6x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-2x^{4}+4x^{3}-6x^{2};}\phantom{;}x^{3}-10x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+3-;x^n;}\underline{-x^{3}+2x^{2}-3x\phantom{;}\phantom{-;x^n}}\\\phantom{;-x^{3}+2x^{2}-3x\phantom{;}-;x^n;}-8x^{2}-3x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+3-;x^n-;x^n;}\underline{\phantom{;}8x^{2}-16x\phantom{;}+24\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}8x^{2}-16x\phantom{;}+24\phantom{;}\phantom{;}-;x^n-;x^n;}-19x\phantom{;}+23\phantom{;}\phantom{;}\\\end{array}$

$2x^{2}+x-8+\frac{-19x+23}{x^2-2x+3}$