$$\frac{1}{x+1}+\frac{3}{5x+1} = \frac{5}{x+4}$$
Answer
$$ \begin{matrix}x_1 = 1 & x_2 = -\dfrac{ 11 }{ 17 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x+1}+\frac{3}{5x+1} &= \frac{5}{x+4}&& \text{multiply ALL terms by } \color{blue}{ (x+1)(5x+1)(x+4) }. \\[1 em](x+1)(5x+1)(x+4)\cdot\frac{1}{x+1}+(x+1)(5x+1)(x+4)\cdot\frac{3}{5x+1} &= (x+1)(5x+1)(x+4)\cdot\frac{5}{x+4}&& \text{cancel out the denominators} \\[1 em]5x^2+21x+4+3x^2+15x+12 &= 25x^2+30x+5&& \text{simplify left side} \\[1 em]8x^2+36x+16 &= 25x^2+30x+5&& \text{move all terms to the left hand side } \\[1 em]8x^2+36x+16-25x^2-30x-5 &= 0&& \text{simplify left side} \\[1 em]-17x^2+6x+11 &= 0&& \\[1 em] \end{aligned} $$
$ -17x^{2}+6x+11 = 0 $ is a quadratic equation.
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