Final answer to the problem
$x^{3}+x^{2}+2x+2+\frac{3}{x-1}$
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1
Divide $x^4+x^2+1$ by $x-1$
$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-1;}{\phantom{;}x^{3}+x^{2}+2x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-1;}\underline{-x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}+x^{3};}\phantom{;}x^{3}+x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n;}\underline{-x^{3}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{3}+x^{2}-;x^n;}\phantom{;}2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n;}\underline{-2x^{2}+2x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-2x^{2}+2x\phantom{;}-;x^n-;x^n;}\phantom{;}2x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n;}\underline{-2x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;;;-2x\phantom{;}+2\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}3\phantom{;}\phantom{;}\\\end{array}$
2
Resulting polynomial
$x^{3}+x^{2}+2x+2+\frac{3}{x-1}$
Final answer to the problem
$x^{3}+x^{2}+2x+2+\frac{3}{x-1}$
