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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{x}{x-2} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{x}{x-2} \xlongequal{\text{Step 2}} \frac{ 2 \cdot x }{ 1 \cdot \left( x-2 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2x }{ x-2 } \end{aligned} $$

Multiply $ \dfrac{2x}{x-2} $ by $ x+3 $ to get $ \dfrac{ 2x^2+6x }{ x-2 } $.

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.

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

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

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

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