Multiply $4$ by $ \dfrac{z}{z^2-7z-3} $ to get $ \dfrac{ 4z }{ z^2-7z-3 } $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{z}{z^2-7z-3} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{z}{z^2-7z-3} \xlongequal{\text{Step 2}} \frac{ 4 \cdot z }{ 1 \cdot \left( z^2-7z-3 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4z }{ z^2-7z-3 } \end{aligned} $$

Add $9$ and $ \dfrac{4z}{z^2-7z-3} $ to get $ \dfrac{ \color{purple}{ 9z^2-59z-27 } }{ z^2-7z-3 }$.

Step 1: Write $ 9 $ 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 $\color{blue}{ z^2-7z-3 }$.

$$ \begin{aligned} 9+ \frac{4z}{z^2-7z-3} & \xlongequal{\text{Step 1}} \frac{9}{\color{red}{1}} + \frac{4z}{z^2-7z-3} = \frac{ 9 \cdot \color{blue}{ \left( z^2-7z-3 \right) }}{ 1 \cdot \color{blue}{ \left( z^2-7z-3 \right) }} + \frac{ 4z }{ z^2-7z-3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9z^2-63z-27 } }{ z^2-7z-3 } + \frac{ \color{purple}{ 4z } }{ z^2-7z-3 }=\frac{ \color{purple}{ 9z^2-59z-27 } }{ z^2-7z-3 } \end{aligned} $$