$$x^3(x+2)(x+2) = 17$$
Answer
$$ \begin{matrix}x_1 = 1.18724 & x_2 = -0.12493+1.44277i & x_3 = -0.12493-1.44277i \\[1 em] x_4 = -2.46869+0.85628i & x_5 = -2.46869-0.85628i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x^3(x+2)(x+2) &= 17&& \text{simplify left side} \\[1 em](x^4+2x^3)(x+2) &= 17&& \\[1 em]x^5+2x^4+2x^4+4x^3 &= 17&& \\[1 em]x^5+4x^4+4x^3 &= 17&& \text{move all terms to the left hand side } \\[1 em]x^5+4x^4+4x^3-17 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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