Subtract $ \dfrac{4}{z} $ from $ 9z $ to get $ \dfrac{ \color{purple}{ 9z^2-4 } }{ z }$.

Step 1: Write $ 9z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ z }$.

$$ \begin{aligned} 9z- \frac{4}{z} & \xlongequal{\text{Step 1}} \frac{9z}{\color{red}{1}} - \frac{4}{z} = \frac{ 9z \cdot \color{blue}{ z }}{ 1 \cdot \color{blue}{ z }} - \frac{ 4 }{ z } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9z^2 } }{ z } - \frac{ \color{purple}{ 4 } }{ z }=\frac{ \color{purple}{ 9z^2-4 } }{ z } \end{aligned} $$

Subtract $1$ from $ \dfrac{9z^2-4}{z} $ to get $ \dfrac{ \color{purple}{ 9z^2-z-4 } }{ z }$.

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

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{z}$.

$$ \begin{aligned} \frac{9z^2-4}{z} -1 & \xlongequal{\text{Step 1}} \frac{9z^2-4}{z} - \frac{1}{\color{red}{1}} = \frac{ 9z^2-4 }{ z } - \frac{ 1 \cdot \color{blue}{ z }}{ 1 \cdot \color{blue}{ z }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9z^2-4 } }{ z } - \frac{ \color{purple}{ z } }{ z }=\frac{ \color{purple}{ 9z^2-z-4 } }{ z } \end{aligned} $$