Final answer to the problem
$x^{3}-x^{2}+4x-4+\frac{8}{x+1}$
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Step-by-step Solution
1
Divide $x^4+3x^2+4$ by $x+1$
$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+1;}{\phantom{;}x^{3}-x^{2}+4x\phantom{;}-4\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}+3x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+1;}\underline{-x^{4}-x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-x^{3};}-x^{3}+3x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n;}\underline{\phantom{;}x^{3}+x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{3}+x^{2}-;x^n;}\phantom{;}4x^{2}\phantom{-;x^n}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n;}\underline{-4x^{2}-4x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-4x^{2}-4x\phantom{;}-;x^n-;x^n;}-4x\phantom{;}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{\phantom{;}4x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}4x\phantom{;}+4\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}8\phantom{;}\phantom{;}\\\end{array}$
$x^{3}-x^{2}+4x-4+\frac{8}{x+1}$
Final answer to the problem
$x^{3}-x^{2}+4x-4+\frac{8}{x+1}$