Divide both the top and bottom numbers by $ \color{orangered}{ 2 } $.

Multiply $ \dfrac{3}{2} $ by $ x $ to get $ \dfrac{ 3x }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{3}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 3x }{ 2 } \end{aligned} $$

Add $x^2+5x$ and $ \dfrac{3x}{2} $ to get $ \dfrac{ \color{purple}{ 2x^2+13x } }{ 2 }$.

Step 1: Write $ x^2+5x $ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} x^2+5x+ \frac{3x}{2} & \xlongequal{\text{Step 1}} \frac{x^2+5x}{\color{red}{1}} + \frac{3x}{2} = \frac{ \left( x^2+5x \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 3x }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^2+10x } }{ 2 } + \frac{ \color{purple}{ 3x } }{ 2 }=\frac{ \color{purple}{ 2x^2+13x } }{ 2 } \end{aligned} $$

Add $ \dfrac{2x^2+13x}{2} $ and $ 3 $ to get $ \dfrac{ \color{purple}{ 2x^2+13x+6 } }{ 2 }$.

Step 1: Write $ 3 $ 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 second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{2x^2+13x}{2} +3 & \xlongequal{\text{Step 1}} \frac{2x^2+13x}{2} + \frac{3}{\color{red}{1}} = \frac{ 2x^2+13x }{ 2 } + \frac{ 3 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^2+13x } }{ 2 } + \frac{ \color{purple}{ 6 } }{ 2 }=\frac{ \color{purple}{ 2x^2+13x+6 } }{ 2 } \end{aligned} $$