Question
$$x^3+6x+\frac{1}{x} = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x^3+6x+\frac{1}{x} &= 0&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]xx^3+x\cdot6x+x\cdot\frac{1}{x} &= x\cdot0&& \text{cancel out the denominators} \\[1 em]x^4+6x^2+1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using quartic formulas
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Equations Solver