Multiply $4$ by $ \dfrac{x}{3} $ to get $ \dfrac{ 4x }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{x}{3} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{x}{3} \xlongequal{\text{Step 2}} \frac{ 4 \cdot x }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ 4x }{ 3 } \end{aligned} $$

Multiply $ \dfrac{4x}{3} $ by $ x^2 $ to get $ \dfrac{ 4x^3 }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{4x^3}{3} $ and $ 4x $ to get $ \dfrac{ \color{purple}{ 4x^3+12x } }{ 3 }$.

Step 1: Write $ 4x $ 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 $\color{blue}{3}$.

$$ \begin{aligned} \frac{4x^3}{3} +4x & \xlongequal{\text{Step 1}} \frac{4x^3}{3} + \frac{4x}{\color{red}{1}} = \frac{ 4x^3 }{ 3 } + \frac{ 4x \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^3 } }{ 3 } + \frac{ \color{purple}{ 12x } }{ 3 }=\frac{ \color{purple}{ 4x^3+12x } }{ 3 } \end{aligned} $$

Add $ \dfrac{4x^3+12x}{3} $ and $ 8 $ to get $ \dfrac{ \color{purple}{ 4x^3+12x+24 } }{ 3 }$.

Step 1: Write $ 8 $ 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 $\color{blue}{3}$.

$$ \begin{aligned} \frac{4x^3+12x}{3} +8 & \xlongequal{\text{Step 1}} \frac{4x^3+12x}{3} + \frac{8}{\color{red}{1}} = \frac{ 4x^3+12x }{ 3 } + \frac{ 8 \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^3+12x } }{ 3 } + \frac{ \color{purple}{ 24 } }{ 3 }=\frac{ \color{purple}{ 4x^3+12x+24 } }{ 3 } \end{aligned} $$