Multiply $6$ by $ \dfrac{x^2}{3} $ to get $ \dfrac{ 6x^2 }{ 3 } $.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 6 \cdot \frac{x^2}{3} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} \cdot \frac{x^2}{3} \xlongequal{\text{Step 2}} \frac{ 6 \cdot x^2 }{ 1 \cdot 3 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6x^2 }{ 3 } \end{aligned} $$

Multiply $ \dfrac{6x^2}{3} $ by $ x^2 $ to get $ \dfrac{ 6x^4 }{ 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.

$$ \begin{aligned} \frac{6x^2}{3} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{6x^2}{3} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 6x^2 \cdot x^2 }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6x^4 }{ 3 } \end{aligned} $$

Add $3x^3$ and $ \dfrac{6x^4}{3} $ to get $ \dfrac{ \color{purple}{ 6x^4+9x^3 } }{ 3 }$.

Step 1: Write $ 3x^3 $ 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 first fraction by $\color{blue}{ 3 }$.

$$ \begin{aligned} 3x^3+ \frac{6x^4}{3} & \xlongequal{\text{Step 1}} \frac{3x^3}{\color{red}{1}} + \frac{6x^4}{3} = \frac{ 3x^3 \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} + \frac{ 6x^4 }{ 3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9x^3 } }{ 3 } + \frac{ \color{purple}{ 6x^4 } }{ 3 }=\frac{ \color{purple}{ 6x^4+9x^3 } }{ 3 } \end{aligned} $$