Final Answer
$x^{4}+x^{3}+3x^{2}-6+\frac{-8x+18}{x^2-x+3}$
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Step-by-step Solution
Specify the solving method
1
Divide $x^6+5x^4+3x^2-2x$ by $x^2-x+3$
$\begin{array}{l}\phantom{\phantom{;}x^{2}-x\phantom{;}+3;}{\phantom{;}x^{4}+x^{3}+3x^{2}\phantom{-;x^n}-6\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-x\phantom{;}+3\overline{\smash{)}\phantom{;}x^{6}\phantom{-;x^n}+5x^{4}\phantom{-;x^n}+3x^{2}-2x\phantom{;}\phantom{-;x^n}}\\\phantom{\phantom{;}x^{2}-x\phantom{;}+3;}\underline{-x^{6}+x^{5}-3x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{6}+x^{5}-3x^{4};}\phantom{;}x^{5}+2x^{4}\phantom{-;x^n}+3x^{2}-2x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}-x\phantom{;}+3-;x^n;}\underline{-x^{5}+x^{4}-3x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{5}+x^{4}-3x^{3}-;x^n;}\phantom{;}3x^{4}-3x^{3}+3x^{2}-2x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}-x\phantom{;}+3-;x^n-;x^n;}\underline{-3x^{4}+3x^{3}-9x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-3x^{4}+3x^{3}-9x^{2}-;x^n-;x^n;}-6x^{2}-2x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}-x\phantom{;}+3-;x^n-;x^n-;x^n;}\underline{\phantom{;}6x^{2}-6x\phantom{;}+18\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}6x^{2}-6x\phantom{;}+18\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}-8x\phantom{;}+18\phantom{;}\phantom{;}\\\end{array}$
2
Resulting polynomial
$x^{4}+x^{3}+3x^{2}-6+\frac{-8x+18}{x^2-x+3}$
Final Answer
$x^{4}+x^{3}+3x^{2}-6+\frac{-8x+18}{x^2-x+3}$