Question
$$x^3+2x^2-29x+\frac{42}{x^3-2x^2-75x+216} = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x^3+2x^2-29x+\frac{42}{x^3-2x^2-75x+216} &= 0&& \text{multiply ALL terms by } \color{blue}{ x^3-2x^2-75x+216 }. \\[1 em](x^3-2x^2-75x+216)x^3+(x^3-2x^2-75x+216)\cdot2x^2-(x^3-2x^2-75x+216)\cdot29x+(x^3-2x^2-75x+216)\cdot\frac{42}{x^3-2x^2-75x+216} &= (x^3-2x^2-75x+216)\cdot0&& \text{cancel out the denominators} \\[1 em]x^6-2x^5-75x^4+216x^3+2x^5-4x^4-150x^3+432x^2-(29x^4-58x^3-2175x^2+6264x)+42 &= 0&& \text{simplify left side} \\[1 em]x^6-79x^4+66x^3+432x^2-(29x^4-58x^3-2175x^2+6264x)+42 &= 0&& \\[1 em]x^6-79x^4+66x^3+432x^2-29x^4+58x^3+2175x^2-6264x+42 &= 0&& \\[1 em]x^6-108x^4+124x^3+2607x^2-6264x+42 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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