$$2-\frac{1}{x}(x+1) = \frac{3}{x+1}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 3 }{ 2 }-\dfrac{\sqrt{ 13 }}{ 2 } & x_2 = \dfrac{ 3 }{ 2 }+\dfrac{\sqrt{ 13 }}{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} 2-\frac{1}{x}(x+1) &= \frac{3}{x+1}&& \text{multiply ALL terms by } \color{blue}{ x(x+1) }. \\[1 em]x(x+1)\cdot2-x(x+1)\frac{1}{x}(x+1) &= x(x+1)\cdot\frac{3}{x+1}&& \text{cancel out the denominators} \\[1 em]2x^2+2x-(x^2+2x+1) &= 3x&& \text{simplify left side} \\[1 em]2x^2+2x-x^2-2x-1 &= 3x&& \\[1 em]2x^2+2x-x^2-2x-1 &= 3x&& \\[1 em]x^2-1 &= 3x&& \text{move all terms to the left hand side } \\[1 em]x^2-1-3x &= 0&& \text{simplify left side} \\[1 em]x^2-3x-1 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}-3x-1 = 0 $ is a quadratic equation.
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