$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}x^{3}-8x^{2}-x\phantom{;}+9\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}x^{5}-8x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-x^{5}\phantom{-;x^n}-x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{3};}-8x^{4}-x^{3}+x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{\phantom{;}8x^{4}\phantom{-;x^n}+8x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}8x^{4}+8x^{2}-;x^n;}-x^{3}+9x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n;}\underline{\phantom{;}x^{3}\phantom{-;x^n}+x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}x^{3}+x\phantom{;}-;x^n-;x^n;}\phantom{;}9x^{2}+x\phantom{;}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n-;x^n;}\underline{-9x^{2}\phantom{-;x^n}-9\phantom{;}\phantom{;}}\\\phantom{;;;-9x^{2}-9\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}x\phantom{;}-11\phantom{;}\phantom{;}\\\end{array}$

$x^{3}-8x^{2}-x+9+\frac{x-11}{x^2+1}$