$\begin{array}{l}\phantom{\phantom{;}x^{2}+4x\phantom{;}+1;}{\phantom{;}x^{2}-4x\phantom{;}+15\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+4x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+4x\phantom{;}+1;}\underline{-x^{4}-4x^{3}-x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-4x^{3}-x^{2};}-4x^{3}-x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+4x\phantom{;}+1-;x^n;}\underline{\phantom{;}4x^{3}+16x^{2}+4x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}4x^{3}+16x^{2}+4x\phantom{;}-;x^n;}\phantom{;}15x^{2}+4x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+4x\phantom{;}+1-;x^n-;x^n;}\underline{-15x^{2}-60x\phantom{;}-15\phantom{;}\phantom{;}}\\\phantom{;;-15x^{2}-60x\phantom{;}-15\phantom{;}\phantom{;}-;x^n-;x^n;}-56x\phantom{;}-13\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-4x+15+\frac{-56x-13}{x^2+4x+1}$