$$\frac{3x^2+24x+48}{x^2}+\frac{x-6}{2x^2} = \frac{1}{x^2}$$
Answer
$$ \begin{matrix}x_1 = -1.48162 & x_2 = 3.28646 & x_3 = 5.36838 \\[1 em] x_4 = -0.58661+1.8033i & x_5 = -0.58661-1.8033i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{3x^2+24x+48}{x^2}+\frac{x-6}{2x^2} &= \frac{1}{x^2}&& \text{multiply ALL terms by } \color{blue}{ x^2\cdot2 }. \\[1 em]x^2\cdot2 \cdot \frac{3x^2+24x+48}{x^2}+x^2\cdot2\frac{x-6}{2x^2} &= x^2\cdot2\cdot\frac{1}{x^2}&& \text{cancel out the denominators} \\[1 em]6x^2+48x+96+x^5-6x^4 &= 2&& \text{simplify left side} \\[1 em]x^5-6x^4+6x^2+48x+96 &= 2&& \text{move all terms to the left hand side } \\[1 em]x^5-6x^4+6x^2+48x+96-2 &= 0&& \text{simplify left side} \\[1 em]x^5-6x^4+6x^2+48x+94 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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