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

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

Multiply $ \dfrac{1}{6} $ by $ x^3 $ to get $ \dfrac{ x^3 }{ 6 } $.

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.

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

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}{6} $ from $ \dfrac{3x^2}{4} $ to get $ \dfrac{ \color{purple}{ -2x^3+9x^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}{ 3 }$ and the second by $\color{blue}{ 2 }$.

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

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+9x^2}{12} $ and $ x^2 $ to get $ \dfrac{ \color{purple}{ -2x^3+21x^2 } }{ 12 }$.

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 $\color{blue}{12}$.

$$ \begin{aligned} \frac{-2x^3+9x^2}{12} +x^2 & \xlongequal{\text{Step 1}} \frac{-2x^3+9x^2}{12} + \frac{x^2}{\color{red}{1}} = \frac{ -2x^3+9x^2 }{ 12 } + \frac{ x^2 \cdot \color{blue}{ 12 }}{ 1 \cdot \color{blue}{ 12 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -2x^3+9x^2 } }{ 12 } + \frac{ \color{purple}{ 12x^2 } }{ 12 }=\frac{ \color{purple}{ -2x^3+21x^2 } }{ 12 } \end{aligned} $$

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+21x^2}{12} $ to get $ \dfrac{ \color{purple}{ -5x^3+21x^2 } }{ 12 }$.

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

$$ \begin{aligned} \frac{-2x^3+21x^2}{12} - \frac{x^3}{4} & = \frac{ -2x^3+21x^2 }{ 12 } - \frac{ x^3 \cdot \color{blue}{ 3 }}{ 4 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ -2x^3+21x^2 } }{ 12 } - \frac{ \color{purple}{ 3x^3 } }{ 12 }=\frac{ \color{purple}{ -5x^3+21x^2 } }{ 12 } \end{aligned} $$

Add $ \dfrac{-5x^3+21x^2}{12} $ and $ 7 $ to get $ \dfrac{ \color{purple}{ -5x^3+21x^2+84 } }{ 12 }$.

Step 1: Write $ 7 $ 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 $\color{blue}{12}$.

$$ \begin{aligned} \frac{-5x^3+21x^2}{12} +7 & \xlongequal{\text{Step 1}} \frac{-5x^3+21x^2}{12} + \frac{7}{\color{red}{1}} = \frac{ -5x^3+21x^2 }{ 12 } + \frac{ 7 \cdot \color{blue}{ 12 }}{ 1 \cdot \color{blue}{ 12 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -5x^3+21x^2 } }{ 12 } + \frac{ \color{purple}{ 84 } }{ 12 }=\frac{ \color{purple}{ -5x^3+21x^2+84 } }{ 12 } \end{aligned} $$