$$\frac{x+6}{x+3} = \frac{2}{1}-\frac{5x+12}{x+3}$$

$$ \begin{aligned} \frac{x+6}{x+3} &= \frac{2}{1}-\frac{5x+12}{x+3}&& \text{multiply ALL terms by } \color{blue}{ (x+3)\cdot1 }. \\[1 em](x+3)\cdot1 \cdot \frac{x+6}{x+3} &= (x+3)\cdot1\cdot\frac{2}{1}-(x+3)\cdot1\frac{5x+12}{x+3}&& \text{cancel out the denominators} \\[1 em]x+6 &= 2x+6-(5x+12)&& \text{simplify right side} \\[1 em]x+6 &= 2x+6-5x-12&& \\[1 em]x+6 &= -3x-6&& \text{move the $ \color{blue}{ -3x } $ to the left side and $ \color{blue}{ 6 }$ to the right} \\[1 em]x+3x &= -6-6&& \text{simplify left and right hand side} \\[1 em]4x &= -12&& \text{ divide both sides by $ 4 $ } \\[1 em]x &= -\frac{12}{4}&& \\[1 em]x &= -3&& \\[1 em] \end{aligned} $$

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