Multiply $ \dfrac{a+8}{2} $ by $ a $ to get $ \dfrac{ a^2+8a }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{a+8}{2} \cdot a & \xlongequal{\text{Step 1}} \frac{a+8}{2} \cdot \frac{a}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( a+8 \right) \cdot a }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ a^2+8a }{ 2 } \end{aligned} $$

Add $ \dfrac{a^2+8a}{2} $ and $ 16 $ to get $ \dfrac{ \color{purple}{ a^2+8a+32 } }{ 2 }$.

Step 1: Write $ 16 $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{a^2+8a}{2} +16 & \xlongequal{\text{Step 1}} \frac{a^2+8a}{2} + \frac{16}{\color{red}{1}} = \frac{ a^2+8a }{ 2 } + \frac{ 16 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ a^2+8a } }{ 2 } + \frac{ \color{purple}{ 32 } }{ 2 }=\frac{ \color{purple}{ a^2+8a+32 } }{ 2 } \end{aligned} $$