$$ \begin{aligned} 3 \cdot \frac{x+3}{x+1}+2 &= \frac{3x+1}{x+1}&& \text{multiply ALL terms by } \color{blue}{ x+1 }. \\[1 em](x+1)\cdot3 \cdot \frac{x+3}{x+1}+(x+1)\cdot2 &= (x+1)\frac{3x+1}{x+1}&& \text{cancel out the denominators} \\[1 em]3x+9+2x+2 &= 3x+1&& \text{simplify left side} \\[1 em]5x+11 &= 3x+1&& \text{move the $ \color{blue}{ 3x } $ to the left side and $ \color{blue}{ 11 }$ to the right} \\[1 em]5x-3x &= 1-11&& \text{simplify left and right hand side} \\[1 em]2x &= -10&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= -\frac{10}{2}&& \\[1 em]x &= -5&& \\[1 em] \end{aligned} $$

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