$$ \begin{aligned} 6+\frac{12}{x}+3 &= 0&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot6+x\cdot\frac{12}{x}+x\cdot3 &= x\cdot0&& \text{cancel out the denominators} \\[1 em]6x+12+3x &= 0&& \text{simplify left side} \\[1 em]9x+12 &= 0&& \text{ move the constants to the right } \\[1 em]9x &= -12&& \text{ divide both sides by $ 9 $ } \\[1 em]x &= -\frac{12}{9}&& \\[1 em]x &= -\frac{4}{3}&& \\[1 em] \end{aligned} $$

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