$\begin{array}{l}\phantom{\phantom{;}x^{2}+x\phantom{;}+2;}{\phantom{;}x^{3}-x^{2}-x\phantom{;}+3\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+x\phantom{;}+2\overline{\smash{)}\phantom{;}x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+12\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2;}\underline{-x^{5}-x^{4}-2x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{4}-2x^{3};}-x^{4}-2x^{3}\phantom{-;x^n}\phantom{-;x^n}+12\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2-;x^n;}\underline{\phantom{;}x^{4}+x^{3}+2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{4}+x^{3}+2x^{2}-;x^n;}-x^{3}+2x^{2}\phantom{-;x^n}+12\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2-;x^n-;x^n;}\underline{\phantom{;}x^{3}+x^{2}+2x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}x^{3}+x^{2}+2x\phantom{;}-;x^n-;x^n;}\phantom{;}3x^{2}+2x\phantom{;}+12\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{-3x^{2}-3x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{;;;-3x^{2}-3x\phantom{;}-6\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}-x\phantom{;}+6\phantom{;}\phantom{;}\\\end{array}$

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