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

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