$\begin{array}{l}\phantom{\phantom{;}x^{2}-1;}{\phantom{;}x^{6}+5x^{5}+x^{4}+5x^{3}+x^{2}+5x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-1\overline{\smash{)}\phantom{;}x^{8}+5x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+3x^{2}+x\phantom{;}+8\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-1;}\underline{-x^{8}\phantom{-;x^n}+x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{8}+x^{6};}\phantom{;}5x^{7}+x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+3x^{2}+x\phantom{;}+8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-1-;x^n;}\underline{-5x^{7}\phantom{-;x^n}+5x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-5x^{7}+5x^{5}-;x^n;}\phantom{;}x^{6}+5x^{5}\phantom{-;x^n}\phantom{-;x^n}+3x^{2}+x\phantom{;}+8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-1-;x^n-;x^n;}\underline{-x^{6}\phantom{-;x^n}+x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-x^{6}+x^{4}-;x^n-;x^n;}\phantom{;}5x^{5}+x^{4}\phantom{-;x^n}+3x^{2}+x\phantom{;}+8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-1-;x^n-;x^n-;x^n;}\underline{-5x^{5}\phantom{-;x^n}+5x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-5x^{5}+5x^{3}-;x^n-;x^n-;x^n;}\phantom{;}x^{4}+5x^{3}+3x^{2}+x\phantom{;}+8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-1-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;-x^{4}+x^{2}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}5x^{3}+4x^{2}+x\phantom{;}+8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-1-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-5x^{3}\phantom{-;x^n}+5x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;-5x^{3}+5x\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}4x^{2}+6x\phantom{;}+8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-4x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}}\\\phantom{;;;;;;-4x^{2}+4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}6x\phantom{;}+12\phantom{;}\phantom{;}\\\end{array}$

$x^{6}+5x^{5}+x^{4}+5x^{3}+x^{2}+5x+4+\frac{6x+12}{x^2-1}$