$$\frac{x+5}{x+6}-\frac{x+1}{x+2} = \frac{4}{21}$$
Answer
$$ \begin{matrix}x_1 = 1 & x_2 = -9 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x+5}{x+6}-\frac{x+1}{x+2} &= \frac{4}{21}&& \text{multiply ALL terms by } \color{blue}{ (x+6)(x+2)\cdot21 }. \\[1 em](x+6)(x+2)\cdot21 \cdot \frac{x+5}{x+6}-(x+6)(x+2)\cdot21\frac{x+1}{x+2} &= (x+6)(x+2)\cdot21\cdot\frac{4}{21}&& \text{cancel out the denominators} \\[1 em]21x^2+147x+210-(21x^2+147x+126) &= 4x^2+32x+48&& \text{simplify left side} \\[1 em]21x^2+147x+210-21x^2-147x-126 &= 4x^2+32x+48&& \\[1 em]21x^2+147x+210-21x^2-147x-126 &= 4x^2+32x+48&& \\[1 em]84 &= 4x^2+32x+48&& \text{move all terms to the left hand side } \\[1 em]84-4x^2-32x-48 &= 0&& \text{simplify left side} \\[1 em]-4x^2-32x+36 &= 0&& \\[1 em] \end{aligned} $$
$ -4x^{2}-32x+36 = 0 $ is a quadratic equation.
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