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

$x^{2}-4x-1+\frac{8}{x-1}$