Multiply $6$ by $ \dfrac{x}{x^2-4} $ to get $ \dfrac{ 6x }{ x^2-4 } $.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x-2}$.

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

Simplify $ \dfrac{3x+6}{x^2-4} $ to $ \dfrac{3}{x-2} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+2}$.

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