Multiply $ \dfrac{4z+12}{2} $ by $ z $ to get $ \dfrac{ 4z^2+12z }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4z+12}{2} \cdot z & \xlongequal{\text{Step 1}} \frac{4z+12}{2} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 4z+12 \right) \cdot z }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4z^2+12z }{ 2 } \end{aligned} $$

Add $ \dfrac{4z^2+12z}{2} $ and $ 6 $ to get $ \dfrac{ \color{purple}{ 4z^2+12z+12 } }{ 2 }$.

Step 1: Write $ 6 $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{4z^2+12z}{2} +6 & \xlongequal{\text{Step 1}} \frac{4z^2+12z}{2} + \frac{6}{\color{red}{1}} = \frac{ 4z^2+12z }{ 2 } + \frac{ 6 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4z^2+12z } }{ 2 } + \frac{ \color{purple}{ 12 } }{ 2 }=\frac{ \color{purple}{ 4z^2+12z+12 } }{ 2 } \end{aligned} $$