Subtract $ \dfrac{3}{z+9} $ from $ z-7 $ to get $ \dfrac{ \color{purple}{ z^2+2z-66 } }{ z+9 }$.

Step 1: Write $ z-7 $ 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 $\color{blue}{ z+9 }$.

$$ \begin{aligned} z-7- \frac{3}{z+9} & \xlongequal{\text{Step 1}} \frac{z-7}{\color{red}{1}} - \frac{3}{z+9} = \frac{ \left( z-7 \right) \cdot \color{blue}{ \left( z+9 \right) }}{ 1 \cdot \color{blue}{ \left( z+9 \right) }} - \frac{ 3 }{ z+9 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^2+9z-7z-63 } }{ z+9 } - \frac{ \color{purple}{ 3 } }{ z+9 }=\frac{ \color{purple}{ z^2+2z-66 } }{ z+9 } \end{aligned} $$