Multiply $s$ by $ \dfrac{t}{2} $ to get $ \dfrac{ st }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} s \cdot \frac{t}{2} & \xlongequal{\text{Step 1}} \frac{s}{\color{red}{1}} \cdot \frac{t}{2} \xlongequal{\text{Step 2}} \frac{ s \cdot t }{ 1 \cdot 2 } \xlongequal{\text{Step 3}} \frac{ st }{ 2 } \end{aligned} $$

Multiply $4s$ by $ \dfrac{t^3}{3} $ to get $ \dfrac{ 4st^3 }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4s \cdot \frac{t^3}{3} & \xlongequal{\text{Step 1}} \frac{4s}{\color{red}{1}} \cdot \frac{t^3}{3} \xlongequal{\text{Step 2}} \frac{ 4s \cdot t^3 }{ 1 \cdot 3 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4st^3 }{ 3 } \end{aligned} $$

Add $ \dfrac{st}{2} $ and $ \dfrac{4st^3}{3} $ to get $ \dfrac{ \color{purple}{ 8st^3+3st } }{ 6 }$.

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}{ 3 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{st}{2} + \frac{4st^3}{3} & = \frac{ st \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} + \frac{ 4st^3 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3st } }{ 6 } + \frac{ \color{purple}{ 8st^3 } }{ 6 }=\frac{ \color{purple}{ 8st^3+3st } }{ 6 } \end{aligned} $$