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

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