Final answer to the problem
$x^{3}-x^{2}-x+2+\frac{x-4}{x^2+1}$
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1
Divide $x^5-x^4+x^2-2$ by $x^2+1$
$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}x^{3}-x^{2}-x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}x^{5}-x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-x^{5}\phantom{-;x^n}-x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{3};}-x^{4}-x^{3}+x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{\phantom{;}x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{4}+x^{2}-;x^n;}-x^{3}+2x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n;}\underline{\phantom{;}x^{3}\phantom{-;x^n}+x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}x^{3}+x\phantom{;}-;x^n-;x^n;}\phantom{;}2x^{2}+x\phantom{;}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n-;x^n;}\underline{-2x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{;;;-2x^{2}-2\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}x\phantom{;}-4\phantom{;}\phantom{;}\\\end{array}$
2
Resulting polynomial
$x^{3}-x^{2}-x+2+\frac{x-4}{x^2+1}$
Final answer to the problem
$x^{3}-x^{2}-x+2+\frac{x-4}{x^2+1}$
