$$\frac{1+x}{2}+\frac{3-x}{4} = \frac{x}{x^2-4}$$
Answer
$$ \begin{matrix}x_1 = 2.29295 & x_2 = -1.50769 & x_3 = -5.78526 \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= \frac{x}{x^2-4}&& \text{multiply ALL terms by } \color{blue}{ 2\cdot4(x^2-4) }. \\[1 em]2\cdot4(x^2-4)\frac{1+x}{2}+2\cdot4(x^2-4)\frac{3-x}{4} &= 2\cdot4(x^2-4)\frac{x}{x^2-4}&& \text{cancel out the denominators} \\[1 em]4x^3+4x^2-16x-16-2x^3+6x^2+8x-24 &= 8x&& \text{simplify left side} \\[1 em]2x^3+10x^2-8x-40 &= 8x&& \text{move all terms to the left hand side } \\[1 em]2x^3+10x^2-8x-40-8x &= 0&& \text{simplify left side} \\[1 em]2x^3+10x^2-16x-40 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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