Add $ \dfrac{x+3}{2} $ and $ \dfrac{1}{x} $ to get $ \dfrac{ \color{purple}{ x^2+3x+2 } }{ 2x }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ x }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{x+3}{2} + \frac{1}{x} & = \frac{ \left( x+3 \right) \cdot \color{blue}{ x }}{ 2 \cdot \color{blue}{ x }} + \frac{ 1 \cdot \color{blue}{ 2 }}{ x \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ x^2+3x } }{ 2x } + \frac{ \color{purple}{ 2 } }{ 2x }=\frac{ \color{purple}{ x^2+3x+2 } }{ 2x } \end{aligned} $$