$$\frac{1}{x+1}+\frac{4}{3} = \frac{1}{x+5}$$
Answer
$$ \begin{matrix}x_1 = -2 & x_2 = -4 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x+1}+\frac{4}{3} &= \frac{1}{x+5}&& \text{multiply ALL terms by } \color{blue}{ (x+1)\cdot3(x+5) }. \\[1 em](x+1)\cdot3(x+5)\cdot\frac{1}{x+1}+(x+1)\cdot3(x+5)\cdot\frac{4}{3} &= (x+1)\cdot3(x+5)\cdot\frac{1}{x+5}&& \text{cancel out the denominators} \\[1 em]3x+15+4x^2+24x+20 &= 3x+3&& \text{simplify left side} \\[1 em]4x^2+27x+35 &= 3x+3&& \text{move all terms to the left hand side } \\[1 em]4x^2+27x+35-3x-3 &= 0&& \text{simplify left side} \\[1 em]4x^2+24x+32 &= 0&& \\[1 em] \end{aligned} $$
$ 4x^{2}+24x+32 = 0 $ is a quadratic equation.
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