$\begin{array}{l}\phantom{\phantom{;}x^{2}+3x\phantom{;}+1;}{\phantom{;}x^{2}-3x\phantom{;}+8\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+1;}\underline{-x^{4}-3x^{3}-x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-3x^{3}-x^{2};}-3x^{3}-x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+1-;x^n;}\underline{\phantom{;}3x^{3}+9x^{2}+3x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}3x^{3}+9x^{2}+3x\phantom{;}-;x^n;}\phantom{;}8x^{2}+3x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+1-;x^n-;x^n;}\underline{-8x^{2}-24x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{;;-8x^{2}-24x\phantom{;}-8\phantom{;}\phantom{;}-;x^n-;x^n;}-21x\phantom{;}-6\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-3x+8+\frac{-21x-6}{x^2+3x+1}$