Add $z$ and $ \dfrac{4}{z^2} $ to get $ \dfrac{ \color{purple}{ z^3+4 } }{ z^2 }$.

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 first fraction by $\color{blue}{ z^2 }$.

$$ \begin{aligned} z+ \frac{4}{z^2} & \xlongequal{\text{Step 1}} \frac{z}{\color{red}{1}} + \frac{4}{z^2} = \frac{ z \cdot \color{blue}{ z^2 }}{ 1 \cdot \color{blue}{ z^2 }} + \frac{ 4 }{ z^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^3 } }{ z^2 } + \frac{ \color{purple}{ 4 } }{ z^2 }=\frac{ \color{purple}{ z^3+4 } }{ z^2 } \end{aligned} $$

Add $ \dfrac{z^3+4}{z^2} $ and $ 4z $ to get $ \dfrac{ \color{purple}{ 5z^3+4 } }{ z^2 }$.

Step 1: Write $ 4z $ 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 $\color{blue}{z^2}$.

$$ \begin{aligned} \frac{z^3+4}{z^2} +4z & \xlongequal{\text{Step 1}} \frac{z^3+4}{z^2} + \frac{4z}{\color{red}{1}} = \frac{ z^3+4 }{ z^2 } + \frac{ 4z \cdot \color{blue}{ z^2 }}{ 1 \cdot \color{blue}{ z^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^3+4 } }{ z^2 } + \frac{ \color{purple}{ 4z^3 } }{ z^2 }=\frac{ \color{purple}{ 5z^3+4 } }{ z^2 } \end{aligned} $$