$$\frac{x-1}{2x+2}+\frac{x-1}{x+1} = 1$$
Answer
$$ \begin{matrix}x_1 = -1 & x_2 = 5 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x-1}{2x+2}+\frac{x-1}{x+1} &= 1&& \text{multiply ALL terms by } \color{blue}{ (2x+2)(x+1) }. \\[1 em](2x+2)(x+1)\frac{x-1}{2x+2}+(2x+2)(x+1)\frac{x-1}{x+1} &= (2x+2)(x+1)\cdot1&& \text{cancel out the denominators} \\[1 em]x^2-1+2x^2-2 &= 2x^2+4x+2&& \text{simplify left side} \\[1 em]3x^2-3 &= 2x^2+4x+2&& \text{move all terms to the left hand side } \\[1 em]3x^2-3-2x^2-4x-2 &= 0&& \text{simplify left side} \\[1 em]x^2-4x-5 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}-4x-5 = 0 $ is a quadratic equation.
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