$\begin{array}{l}\phantom{\phantom{;}x^{2}+2x\phantom{;}+5;}{\phantom{;}x^{2}-2x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+2x\phantom{;}+5\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}+5;}\underline{-x^{4}-2x^{3}-5x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-2x^{3}-5x^{2};}-2x^{3}-5x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}+5-;x^n;}\underline{\phantom{;}2x^{3}+4x^{2}+10x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}2x^{3}+4x^{2}+10x\phantom{;}-;x^n;}-x^{2}+10x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}+5-;x^n-;x^n;}\underline{\phantom{;}x^{2}+2x\phantom{;}+5\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}x^{2}+2x\phantom{;}+5\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}12x\phantom{;}+7\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-2x-1+\frac{12x+7}{x^2+2x+5}$