Question
$$3x^5+4x^2+6x+2x^5+8x^2+5x = 0$$
Answer
$$ x = 0 $$
Explanation
$$ \begin{aligned} 3x^5+4x^2+6x+2x^5+8x^2+5x &= 0&& \text{simplify left side} \\[1 em]5x^5+12x^2+11x &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ 5x^{5}+12x^{2}+11x = 0 } $, first we need to factor our $ x $.
$$ 5x^{5}+12x^{2}+11x = x \left( 5x^{4}+12x+11 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The remaining roots can be found by solving equation $ 5x^{4}+12x+11 = 0$.
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using quartic formulas
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Polynomial Equations Solver