Question
$$\frac{3}{x+1}-\frac{1}{x+2} = \frac{5}{4}$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 7 }{ 10 }-\dfrac{\sqrt{ 249 }}{ 10 } & x_2 = -\dfrac{ 7 }{ 10 }+\dfrac{\sqrt{ 249 }}{ 10 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{3}{x+1}-\frac{1}{x+2} &= \frac{5}{4}&& \text{multiply ALL terms by } \color{blue}{ (x+1)(x+2)\cdot4 }. \\[1 em](x+1)(x+2)\cdot4\cdot\frac{3}{x+1}-(x+1)(x+2)\cdot4\cdot\frac{1}{x+2} &= (x+1)(x+2)\cdot4\cdot\frac{5}{4}&& \text{cancel out the denominators} \\[1 em]12x+24-(4x+4) &= 5x^2+15x+10&& \text{simplify left side} \\[1 em]12x+24-4x-4 &= 5x^2+15x+10&& \\[1 em]8x+20 &= 5x^2+15x+10&& \text{move all terms to the left hand side } \\[1 em]8x+20-5x^2-15x-10 &= 0&& \text{simplify left side} \\[1 em]-5x^2-7x+10 &= 0&& \\[1 em] \end{aligned} $$
$ -5x^{2}-7x+10 = 0 $ is a quadratic equation.
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