Subtract $ \dfrac{z}{3z+5} $ from $ 4z $ to get $ \dfrac{ \color{purple}{ 12z^2+19z } }{ 3z+5 }$.

Step 1: Write $ 4z $ 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 $\color{blue}{ 3z+5 }$.

$$ \begin{aligned} 4z- \frac{z}{3z+5} & \xlongequal{\text{Step 1}} \frac{4z}{\color{red}{1}} - \frac{z}{3z+5} = \frac{ 4z \cdot \color{blue}{ \left( 3z+5 \right) }}{ 1 \cdot \color{blue}{ \left( 3z+5 \right) }} - \frac{ z }{ 3z+5 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 12z^2+20z } }{ 3z+5 } - \frac{ \color{purple}{ z } }{ 3z+5 }=\frac{ \color{purple}{ 12z^2+19z } }{ 3z+5 } \end{aligned} $$