$$ \begin{aligned} \frac{1}{x+3}+\frac{1}{x+12} &= \frac{1}{x}&& \text{multiply ALL terms by } \color{blue}{ (x+3)(x+12)x }. \\[1 em](x+3)(x+12)x\cdot\frac{1}{x+3}+(x+3)(x+12)x\cdot\frac{1}{x+12} &= (x+3)(x+12)x\cdot\frac{1}{x}&& \text{cancel out the denominators} \\[1 em]x^2+12x+x^2+3x &= x^2+15x+36&& \text{simplify left side} \\[1 em]2x^2+15x &= x^2+15x+36&& \text{move all terms to the left hand side } \\[1 em]2x^2+15x-x^2-15x-36 &= 0&& \text{simplify left side} \\[1 em]2x^2+15x-x^2-15x-36 &= 0&& \\[1 em]x^2-36 &= 0&& \\[1 em] \end{aligned} $$

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