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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{9}{4} $ by $ x^2 $ to get $ \dfrac{ 9x^2 }{ 4 } $.

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.

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

Subtract $ \dfrac{3x}{2} $ from $ 3 $ to get $ \dfrac{ \color{purple}{ -3x+6 } }{ 2 }$.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 2 }$.

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

Add $9-9x$ and $ \dfrac{9x^2}{4} $ to get $ \dfrac{ \color{purple}{ 9x^2-36x+36 } }{ 4 }$.

Step 1: Write $ 9-9x $ 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 $\color{blue}{ 4 }$.

$$ \begin{aligned} 9-9x+ \frac{9x^2}{4} & \xlongequal{\text{Step 1}} \frac{9-9x}{\color{red}{1}} + \frac{9x^2}{4} = \frac{ \left( 9-9x \right) \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} + \frac{ 9x^2 }{ 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 36-36x } }{ 4 } + \frac{ \color{purple}{ 9x^2 } }{ 4 }=\frac{ \color{purple}{ 9x^2-36x+36 } }{ 4 } \end{aligned} $$

Multiply $ \dfrac{-3x+6}{2} $ by $ \dfrac{9x^2-36x+36}{4} $ to get $ \dfrac{-27x^3+162x^2-324x+216}{8} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-3x+6}{2} \cdot \frac{9x^2-36x+36}{4} & \xlongequal{\text{Step 1}} \frac{ \left( -3x+6 \right) \cdot \left( 9x^2-36x+36 \right) }{ 2 \cdot 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ -27x^3+108x^2-108x+54x^2-216x+216 }{ 8 } = \frac{-27x^3+162x^2-324x+216}{8} \end{aligned} $$