$$x^6+5 \cdot \frac{x^4}{4}+\frac{x^2}{4}-1 = 0$$
Answer
$$ \begin{matrix}x_1 = 0.81298 & x_2 = -0.81298 & x_3 = -0.37052+1.04535i \\[1 em] x_4 = -0.37052-1.04535i & x_5 = 0.37052+1.04535i & x_6 = 0.37052-1.04535i \end{matrix} $$
Explanation
$$ \begin{aligned} x^6+5 \cdot \frac{x^4}{4}+\frac{x^2}{4}-1 &= 0&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4x^6+4\cdot5 \cdot \frac{x^4}{4}+4\frac{x^2}{4}-4\cdot1 &= 4\cdot0&& \text{cancel out the denominators} \\[1 em]4x^6+5x^4+x^2-4 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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