$$8x^4+4x^3+\frac{3}{x} = 0$$
Answer
$$ \begin{matrix}x_1 = -0.95356 & x_2 = 0.58253+0.47333i & x_3 = 0.58253-0.47333i \\[1 em] x_4 = -0.35574+0.75597i & x_5 = -0.35574-0.75597i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} 8x^4+4x^3+\frac{3}{x} &= 0&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot8x^4+x\cdot4x^3+x\cdot\frac{3}{x} &= x\cdot0&& \text{cancel out the denominators} \\[1 em]8x^5+4x^4+3 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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Polynomial Equations Solver