Multiply $z+3$ by $ \dfrac{z-4}{z} $ to get $ \dfrac{z^2-z-12}{z} $.

Step 1: Write $ z+3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{z^2-z-12}{z} $ by $ z-3 $ to get $ \dfrac{z^3-4z^2-9z+36}{z} $.

Step 1: Write $ z-3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{z^2-z-12}{z} \cdot z-3 & \xlongequal{\text{Step 1}} \frac{z^2-z-12}{z} \cdot \frac{z-3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( z^2-z-12 \right) \cdot \left( z-3 \right) }{ z \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ z^3-3z^2-z^2+3z-12z+36 }{ z } = \frac{z^3-4z^2-9z+36}{z} \end{aligned} $$