$$\frac{4}{x^2}+5x+6 = \frac{2}{x^2}+6x+8$$
Answer
$$ \begin{matrix}x_1 = 0.83929 & x_2 = -1.41964+0.60629i & x_3 = -1.41964-0.60629i \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{4}{x^2}+5x+6 &= \frac{2}{x^2}+6x+8&& \text{multiply ALL terms by } \color{blue}{ x^2 }. \\[1 em]x^2\cdot\frac{4}{x^2}+x^2\cdot5x+x^2\cdot6 &= x^2\cdot\frac{2}{x^2}+x^2\cdot6x+x^2\cdot8&& \text{cancel out the denominators} \\[1 em]4+5x^3+6x^2 &= 2+6x^3+8x^2&& \text{simplify left and right hand side} \\[1 em]5x^3+6x^2+4 &= 6x^3+8x^2+2&& \text{move all terms to the left hand side } \\[1 em]5x^3+6x^2+4-6x^3-8x^2-2 &= 0&& \text{simplify left side} \\[1 em]-x^3-2x^2+2 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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