$$x^5-x^3 = \frac{3}{2}$$
Answer
$$ \begin{matrix}x_1 = 1.29826 & x_2 = 0.36809+0.87511i & x_3 = 0.36809-0.87511i \\[1 em] x_4 = -1.01722+0.49715i & x_5 = -1.01722-0.49715i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x^5-x^3 &= \frac{3}{2}&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2x^5-2x^3 &= 2\cdot\frac{3}{2}&& \text{cancel out the denominators} \\[1 em]2x^5-2x^3 &= 3&& \text{move all terms to the left hand side } \\[1 em]2x^5-2x^3-3 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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