Question
$$27-3(4x-(x^2+5-x^3))+17x^2 = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} 27-3(4x-(x^2+5-x^3))+17x^2 &= 0&& \text{simplify left side} \\[1 em]27-3(4x-x^2-5+x^3)+17x^2 &= 0&& \\[1 em]27-(12x-3x^2-15+3x^3)+17x^2 &= 0&& \\[1 em]27-12x+3x^2+15-3x^3+17x^2 &= 0&& \\[1 em]-3x^3+20x^2-12x+42 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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