Multiply $9$ by $ \dfrac{x^3}{2} $ to get $ \dfrac{ 9x^3 }{ 2 } $.

Step 1: Write $ 9 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{9x^3}{2} $ by $ x^2 $ to get $ \dfrac{ 9x^5 }{ 2 } $.

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.

$$ \begin{aligned} \frac{9x^3}{2} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{9x^3}{2} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 9x^3 \cdot x^2 }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 9x^5 }{ 2 } \end{aligned} $$

Add $x^4$ and $ \dfrac{9x^5}{2} $ to get $ \dfrac{ \color{purple}{ 9x^5+2x^4 } }{ 2 }$.

Step 1: Write $ x^4 $ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} x^4+ \frac{9x^5}{2} & \xlongequal{\text{Step 1}} \frac{x^4}{\color{red}{1}} + \frac{9x^5}{2} = \frac{ x^4 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 9x^5 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^4 } }{ 2 } + \frac{ \color{purple}{ 9x^5 } }{ 2 }=\frac{ \color{purple}{ 9x^5+2x^4 } }{ 2 } \end{aligned} $$