$$\frac{3}{x}+1 = \frac{12}{x^2+1}+2$$
Answer
$$ \begin{matrix}x_1 = 0.24332 & x_2 = 1.37834+3.22946i & x_3 = 1.37834-3.22946i \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{3}{x}+1 &= \frac{12}{x^2+1}+2&& \text{multiply ALL terms by } \color{blue}{ x(x^2+1) }. \\[1 em]x(x^2+1)\cdot\frac{3}{x}+x(x^2+1)\cdot1 &= x(x^2+1)\cdot\frac{12}{x^2+1}+x(x^2+1)\cdot2&& \text{cancel out the denominators} \\[1 em]3x^2+3+x^3+x &= 12x+2x^3+2x&& \text{simplify left and right hand side} \\[1 em]x^3+3x^2+x+3 &= 2x^3+14x&& \text{move all terms to the left hand side } \\[1 em]x^3+3x^2+x+3-2x^3-14x &= 0&& \text{simplify left side} \\[1 em]-x^3+3x^2-13x+3 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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