$$ \begin{aligned} \frac{x+2}{x+3} &= \frac{x-7}{x+1}&& \text{multiply ALL terms by } \color{blue}{ (x+3)(x+1) }. \\[1 em](x+3)(x+1)\frac{x+2}{x+3} &= (x+3)(x+1)\frac{x-7}{x+1}&& \text{cancel out the denominators} \\[1 em]x^2+3x+2 &= x^2-4x-21&& \text{move all terms to the left hand side } \\[1 em]x^2+3x+2-x^2+4x+21 &= 0&& \text{simplify left side} \\[1 em]x^2+3x+2-x^2+4x+21 &= 0&& \\[1 em]7x+23 &= 0&& \text{ move the constants to the right } \\[1 em]7x &= -23&& \text{ divide both sides by $ 7 $ } \\[1 em]x &= -\frac{23}{7}&& \\[1 em] \end{aligned} $$

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