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{1}{3} $ by $ x^3 $ to get $ \dfrac{ x^3 }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{1}{4} $ by $ x^3 $ to get $ \dfrac{ x^3 }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{x^3}{3} $ from $ \dfrac{x^2}{2} $ to get $ \dfrac{ \color{purple}{ -2x^3+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}{ 3 }$ and the second by $\color{blue}{ 2 }$.

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

Multiply $ \dfrac{1}{4} $ by $ x^3 $ to get $ \dfrac{ x^3 }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

Step 1: Write $ x^2 $ 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 $\color{blue}{6}$.

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

Multiply $ \dfrac{1}{4} $ by $ x^3 $ to get $ \dfrac{ x^3 }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{x^3}{4} $ from $ \dfrac{-2x^3+9x^2}{6} $ to get $ \dfrac{ \color{purple}{ -7x^3+18x^2 } }{ 12 }$.

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 $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{-2x^3+9x^2}{6} - \frac{x^3}{4} & = \frac{ \left( -2x^3+9x^2 \right) \cdot \color{blue}{ 2 }}{ 6 \cdot \color{blue}{ 2 }} - \frac{ x^3 \cdot \color{blue}{ 3 }}{ 4 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ -4x^3+18x^2 } }{ 12 } - \frac{ \color{purple}{ 3x^3 } }{ 12 }=\frac{ \color{purple}{ -7x^3+18x^2 } }{ 12 } \end{aligned} $$

Add $ \dfrac{-7x^3+18x^2}{12} $ and $ 14 $ to get $ \dfrac{ \color{purple}{ -7x^3+18x^2+168 } }{ 12 }$.

Step 1: Write $ 14 $ 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 $\color{blue}{12}$.

$$ \begin{aligned} \frac{-7x^3+18x^2}{12} +14 & \xlongequal{\text{Step 1}} \frac{-7x^3+18x^2}{12} + \frac{14}{\color{red}{1}} = \frac{ -7x^3+18x^2 }{ 12 } + \frac{ 14 \cdot \color{blue}{ 12 }}{ 1 \cdot \color{blue}{ 12 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -7x^3+18x^2 } }{ 12 } + \frac{ \color{purple}{ 168 } }{ 12 }=\frac{ \color{purple}{ -7x^3+18x^2+168 } }{ 12 } \end{aligned} $$