$$\frac{3}{x^2}+\frac{4}{x} = 5$$
Answer
$$ \begin{matrix}x_1 = 1.10349 & x_2 = -0.59419 & x_3 = -0.25465+0.92208i \\[1 em] x_4 = -0.25465-0.92208i & \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{3}{x^2}+\frac{4}{x} &= 5&& \text{multiply ALL terms by } \color{blue}{ x^2x }. \\[1 em]x^2x\cdot\frac{3}{x^2}+x^2x\cdot\frac{4}{x} &= x^2x\cdot5&& \text{cancel out the denominators} \\[1 em]\frac{3}{x^1}+4 &= 5x^3&& \text{multiply ALL terms by } \color{blue}{ x^1 }. \\[1 em]x^1\cdot\frac{3}{x^1}+x^1\cdot4 &= x^1\cdot5x^3&& \text{cancel out the denominators} \\[1 em]3+4x &= 5x^4&& \text{move all terms to the left hand side } \\[1 em]3+4x-5x^4 &= 0&& \text{simplify left side} \\[1 em]-5x^4+4x+3 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using quartic formulas
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