$$ \begin{aligned} \frac{1}{x+3}+\frac{1}{x+12} &= 0&& \text{multiply ALL terms by } \color{blue}{ (x+3)(x+12) }. \\[1 em](x+3)(x+12)\cdot\frac{1}{x+3}+(x+3)(x+12)\cdot\frac{1}{x+12} &= (x+3)(x+12)\cdot0&& \text{cancel out the denominators} \\[1 em]x+12+x+3 &= 0&& \text{simplify left side} \\[1 em]2x+15 &= 0&& \text{ move the constants to the right } \\[1 em]2x &= -15&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= -\frac{15}{2}&& \\[1 em] \end{aligned} $$

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