$$\frac{3}{x+1} = \frac{12}{x^2+1}+2$$
Answer
$$ \begin{matrix}x_1 = -0.70131 & x_2 = 0.60065+2.73527i & x_3 = 0.60065-2.73527i \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{3}{x+1} &= \frac{12}{x^2+1}+2&& \text{multiply ALL terms by } \color{blue}{ (x+1)(x^2+1) }. \\[1 em](x+1)(x^2+1)\cdot\frac{3}{x+1} &= (x+1)(x^2+1)\cdot\frac{12}{x^2+1}+(x+1)(x^2+1)\cdot2&& \text{cancel out the denominators} \\[1 em]3x^2+3 &= 12x+12+2x^3+2x^2+2x+2&& \text{simplify right side} \\[1 em]3x^2+3 &= 2x^3+2x^2+14x+14&& \text{move all terms to the left hand side } \\[1 em]3x^2+3-2x^3-2x^2-14x-14 &= 0&& \text{simplify left side} \\[1 em]-2x^3+x^2-14x-11 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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