Add $ \dfrac{z}{z+5} $ and $ z $ to get $ \dfrac{ \color{purple}{ z^2+6z } }{ z+5 }$.

Step 1: Write $ z $ 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 $\color{blue}{z+5}$.

$$ \begin{aligned} \frac{z}{z+5} +z & \xlongequal{\text{Step 1}} \frac{z}{z+5} + \frac{z}{\color{red}{1}} = \frac{ z }{ z+5 } + \frac{ z \cdot \color{blue}{ \left( z+5 \right) }}{ 1 \cdot \color{blue}{ \left( z+5 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z } }{ z+5 } + \frac{ \color{purple}{ z^2+5z } }{ z+5 }=\frac{ \color{purple}{ z^2+6z } }{ z+5 } \end{aligned} $$