Multiply $10$ by $ \dfrac{z}{z-5} $ to get $ \dfrac{ 10z }{ z-5 } $.

Step 1: Write $ 10 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 10 \cdot \frac{z}{z-5} & \xlongequal{\text{Step 1}} \frac{10}{\color{red}{1}} \cdot \frac{z}{z-5} \xlongequal{\text{Step 2}} \frac{ 10 \cdot z }{ 1 \cdot \left( z-5 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10z }{ z-5 } \end{aligned} $$

Add $ \dfrac{9}{z^2-11z+30} $ and $ \dfrac{10z}{z-5} $ to get $ \dfrac{ \color{purple}{ 10z^2-60z+9 } }{ z^2-11z+30 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{z-6}$.

$$ \begin{aligned} \frac{9}{z^2-11z+30} + \frac{10z}{z-5} & = \frac{ 9 }{ z^2-11z+30 } + \frac{ 10z \cdot \color{blue}{ \left( z-6 \right) }}{ \left( z-5 \right) \cdot \color{blue}{ \left( z-6 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 9 } }{ z^2-11z+30 } + \frac{ \color{purple}{ 10z^2-60z } }{ z^2-11z+30 }=\frac{ \color{purple}{ 10z^2-60z+9 } }{ z^2-11z+30 } \end{aligned} $$