Multiply $12$ by $ \dfrac{x^2}{3} $ to get $ \dfrac{ 12x^2 }{ 3 } $.

Step 1: Write $ 12 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{12x^2}{3} $ by $ x^5 $ to get $ \dfrac{ 12x^7 }{ 3 } $.

Step 1: Write $ x^5 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

Step 1: Write $ 3x^4 $ 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 $\color{blue}{ 3 }$.

$$ \begin{aligned} 3x^4- \frac{12x^7}{3} & \xlongequal{\text{Step 1}} \frac{3x^4}{\color{red}{1}} - \frac{12x^7}{3} = \frac{ 3x^4 \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} - \frac{ 12x^7 }{ 3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9x^4 } }{ 3 } - \frac{ \color{purple}{ 12x^7 } }{ 3 }=\frac{ \color{purple}{ -12x^7+9x^4 } }{ 3 } \end{aligned} $$

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

Step 1: Write $ 3x^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 second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{-12x^7+9x^4}{3} -3x^3 & \xlongequal{\text{Step 1}} \frac{-12x^7+9x^4}{3} - \frac{3x^3}{\color{red}{1}} = \frac{ -12x^7+9x^4 }{ 3 } - \frac{ 3x^3 \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -12x^7+9x^4 } }{ 3 } - \frac{ \color{purple}{ 9x^3 } }{ 3 }=\frac{ \color{purple}{ -12x^7+9x^4-9x^3 } }{ 3 } \end{aligned} $$