$\begin{array}{l}\phantom{\phantom{;}x^{2}+5x\phantom{;}+1;}{\phantom{;}x^{2}-5x\phantom{;}+24\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+5x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+1;}\underline{-x^{4}-5x^{3}-x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-5x^{3}-x^{2};}-5x^{3}-x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+1-;x^n;}\underline{\phantom{;}5x^{3}+25x^{2}+5x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}5x^{3}+25x^{2}+5x\phantom{;}-;x^n;}\phantom{;}24x^{2}+5x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+1-;x^n-;x^n;}\underline{-24x^{2}-120x\phantom{;}-24\phantom{;}\phantom{;}}\\\phantom{;;-24x^{2}-120x\phantom{;}-24\phantom{;}\phantom{;}-;x^n-;x^n;}-115x\phantom{;}-22\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-5x+24+\frac{-115x-22}{x^2+5x+1}$