Multiply $ \dfrac{1}{3} $ by $ a $ to get $ \dfrac{ a }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{1}{2} $ by $ x^2 $ to get $ \dfrac{ x^2 }{ 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.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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.

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

Multiply $ \dfrac{1}{2} $ by $ x^2 $ to get $ \dfrac{ x^2 }{ 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.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

Step 1: Write $ ax $ 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 second fraction by $\color{blue}{2}$.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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

$$ \begin{aligned} \frac{ax}{3} - \frac{x^2}{2} & = \frac{ ax \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} - \frac{ x^2 \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 2ax } }{ 6 } - \frac{ \color{purple}{ 3x^2 } }{ 6 }=\frac{ \color{purple}{ 2ax-3x^2 } }{ 6 } \end{aligned} $$

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{-2ax+3x^2}{2} $ and $ \dfrac{2a^2}{3} $ to get $ \dfrac{ \color{purple}{ 4a^2-6ax+9x^2 } }{ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{-2ax+3x^2}{2} + \frac{2a^2}{3} & = \frac{ \left( -2ax+3x^2 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} + \frac{ 2a^2 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ -6ax+9x^2 } }{ 6 } + \frac{ \color{purple}{ 4a^2 } }{ 6 }=\frac{ \color{purple}{ 4a^2-6ax+9x^2 } }{ 6 } \end{aligned} $$

Add $ \dfrac{2ax-3x^2}{6} $ and $ \dfrac{3a^2}{2} $ to get $ \dfrac{ \color{purple}{ 9a^2+2ax-3x^2 } }{ 6 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{2ax-3x^2}{6} + \frac{3a^2}{2} & = \frac{ 2ax-3x^2 }{ 6 } + \frac{ 3a^2 \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 2ax-3x^2 } }{ 6 } + \frac{ \color{purple}{ 9a^2 } }{ 6 }=\frac{ \color{purple}{ 9a^2+2ax-3x^2 } }{ 6 } \end{aligned} $$

Add $ \dfrac{-2ax+3x^2}{2} $ and $ \dfrac{2a^2}{3} $ to get $ \dfrac{ \color{purple}{ 4a^2-6ax+9x^2 } }{ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{-2ax+3x^2}{2} + \frac{2a^2}{3} & = \frac{ \left( -2ax+3x^2 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} + \frac{ 2a^2 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ -6ax+9x^2 } }{ 6 } + \frac{ \color{purple}{ 4a^2 } }{ 6 }=\frac{ \color{purple}{ 4a^2-6ax+9x^2 } }{ 6 } \end{aligned} $$

Multiply $ \dfrac{9a^2+2ax-3x^2}{6} $ by $ \dfrac{4a^2-6ax+9x^2}{6} $ to get $ \dfrac{36a^4-46a^3x+57a^2x^2+36ax^3-27x^4}{36} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{9a^2+2ax-3x^2}{6} \cdot \frac{4a^2-6ax+9x^2}{6} & \xlongequal{\text{Step 1}} \frac{ \left( 9a^2+2ax-3x^2 \right) \cdot \left( 4a^2-6ax+9x^2 \right) }{ 6 \cdot 6 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 36a^4-54a^3x+81a^2x^2+8a^3x-12a^2x^2+18ax^3-12a^2x^2+18ax^3-27x^4 }{ 36 } = \frac{36a^4-46a^3x+57a^2x^2+36ax^3-27x^4}{36} \end{aligned} $$