$$\frac{x}{x^2+3x+2} = \frac{x-2}{x^2+5x+4}$$
Answer
$$ x = -1 $$
Explanation
$$ \begin{aligned} \frac{x}{x^2+3x+2} &= \frac{x-2}{x^2+5x+4}&& \text{multiply ALL terms by } \color{blue}{ (x^2+3x+2)(x^2+5x+4) }. \\[1 em](x^2+3x+2)(x^2+5x+4)\frac{x}{x^2+3x+2} &= (x^2+3x+2)(x^2+5x+4)\frac{x-2}{x^2+5x+4}&& \text{cancel out the denominators} \\[1 em]x^3+5x^2+4x &= x^3+x^2-4x-4&& \text{move all terms to the left hand side } \\[1 em]x^3+5x^2+4x-x^3-x^2+4x+4 &= 0&& \text{simplify left side} \\[1 em]x^3+5x^2+4x-x^3-x^2+4x+4 &= 0&& \\[1 em]4x^2+8x+4 &= 0&& \\[1 em] \end{aligned} $$
$ 4x^{2}+8x+4 = 0 $ is a quadratic equation.
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