$\begin{array}{l}\phantom{\phantom{;}x^{2}+x\phantom{;}+2;}{\phantom{;}x^{4}-x^{3}-6x^{2}+8x\phantom{;}+7\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+x\phantom{;}+2\overline{\smash{)}\phantom{;}x^{6}\phantom{-;x^n}-5x^{4}\phantom{-;x^n}+3x^{2}-2x\phantom{;}\phantom{-;x^n}}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2;}\underline{-x^{6}-x^{5}-2x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{6}-x^{5}-2x^{4};}-x^{5}-7x^{4}\phantom{-;x^n}+3x^{2}-2x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2-;x^n;}\underline{\phantom{;}x^{5}+x^{4}+2x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{5}+x^{4}+2x^{3}-;x^n;}-6x^{4}+2x^{3}+3x^{2}-2x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2-;x^n-;x^n;}\underline{\phantom{;}6x^{4}+6x^{3}+12x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}6x^{4}+6x^{3}+12x^{2}-;x^n-;x^n;}\phantom{;}8x^{3}+15x^{2}-2x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{-8x^{3}-8x^{2}-16x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-8x^{3}-8x^{2}-16x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}7x^{2}-18x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}+x\phantom{;}+2-;x^n-;x^n-;x^n-;x^n;}\underline{-7x^{2}-7x\phantom{;}-14\phantom{;}\phantom{;}}\\\phantom{;;;;-7x^{2}-7x\phantom{;}-14\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-25x\phantom{;}-14\phantom{;}\phantom{;}\\\end{array}$

$x^{4}-x^{3}-6x^{2}+8x+7+\frac{-25x-14}{x^2+x+2}$