$$ \begin{aligned} \frac{1}{x+2}+2 &= \frac{9}{4}&& \text{multiply ALL terms by } \color{blue}{ (x+2)\cdot4 }. \\[1 em](x+2)\cdot4\cdot\frac{1}{x+2}+(x+2)\cdot4\cdot2 &= (x+2)\cdot4\cdot\frac{9}{4}&& \text{cancel out the denominators} \\[1 em]4+8x+16 &= 9x+18&& \text{simplify left side} \\[1 em]8x+20 &= 9x+18&& \text{move the $ \color{blue}{ 9x } $ to the left side and $ \color{blue}{ 20 }$ to the right} \\[1 em]8x-9x &= 18-20&& \text{simplify left and right hand side} \\[1 em]-x &= -2&& \text{switch signs on both sides} \\[1 em]x &= 2&& \\[1 em] \end{aligned} $$

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