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

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} x+ \frac{1}{2} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} + \frac{1}{2} = \frac{ x \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 1 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x } }{ 2 } + \frac{ \color{purple}{ 1 } }{ 2 }=\frac{ \color{purple}{ 2x+1 } }{ 2 } \end{aligned} $$

Multiply $3x+2$ by $ \dfrac{2x+1}{2} $ to get $ \dfrac{6x^2+7x+2}{2} $.

Step 1: Write $ 3x+2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3x+2 \cdot \frac{2x+1}{2} & \xlongequal{\text{Step 1}} \frac{3x+2}{\color{red}{1}} \cdot \frac{2x+1}{2} \xlongequal{\text{Step 2}} \frac{ \left( 3x+2 \right) \cdot \left( 2x+1 \right) }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6x^2+3x+4x+2 }{ 2 } = \frac{6x^2+7x+2}{2} \end{aligned} $$