Multiply $ \dfrac{2a^2+10a}{3} $ by $ z^2 $ to get $ \dfrac{ 2a^2z^2+10az^2 }{ 3 } $.

Step 1: Write $ z^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2a^2z^2+10az^2}{3} $ and $ 15a $ to get $ \dfrac{ \color{purple}{ 2a^2z^2+10az^2+45a } }{ 3 }$.

Step 1: Write $ 15a $ 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 second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{2a^2z^2+10az^2}{3} +15a & \xlongequal{\text{Step 1}} \frac{2a^2z^2+10az^2}{3} + \frac{15a}{\color{red}{1}} = \frac{ 2a^2z^2+10az^2 }{ 3 } + \frac{ 15a \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2a^2z^2+10az^2 } }{ 3 } + \frac{ \color{purple}{ 45a } }{ 3 }=\frac{ \color{purple}{ 2a^2z^2+10az^2+45a } }{ 3 } \end{aligned} $$