Multiply $12$ by $ \dfrac{x}{6} $ to get $ \dfrac{ 12x }{ 6 } $.

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.

$$ \begin{aligned} 12 \cdot \frac{x}{6} & \xlongequal{\text{Step 1}} \frac{12}{\color{red}{1}} \cdot \frac{x}{6} \xlongequal{\text{Step 2}} \frac{ 12 \cdot x }{ 1 \cdot 6 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 12x }{ 6 } \end{aligned} $$

Multiply $ \dfrac{12x}{6} $ by $ x $ to get $ \dfrac{ 12x^2 }{ 6 } $.

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.

$$ \begin{aligned} \frac{12x}{6} \cdot x & \xlongequal{\text{Step 1}} \frac{12x}{6} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 12x \cdot x }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 12x^2 }{ 6 } \end{aligned} $$

Subtract $ \dfrac{12x^2}{6} $ from $ 11x^2 $ to get $ \dfrac{ \color{purple}{ 54x^2 } }{ 6 }$.

Step 1: Write $ 11x^2 $ 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 $\color{blue}{ 6 }$.

$$ \begin{aligned} 11x^2- \frac{12x^2}{6} & \xlongequal{\text{Step 1}} \frac{11x^2}{\color{red}{1}} - \frac{12x^2}{6} = \frac{ 11x^2 \cdot \color{blue}{ 6 }}{ 1 \cdot \color{blue}{ 6 }} - \frac{ 12x^2 }{ 6 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 66x^2 } }{ 6 } - \frac{ \color{purple}{ 12x^2 } }{ 6 }=\frac{ \color{purple}{ 54x^2 } }{ 6 } \end{aligned} $$