Add $ \dfrac{1}{x^2-4} $ and $ \dfrac{6}{x+2} $ to get $ \dfrac{ \color{purple}{ 6x-11 } }{ x^2-4 }$.

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

$$ \begin{aligned} \frac{1}{x^2-4} + \frac{6}{x+2} & = \frac{ 1 }{ x^2-4 } + \frac{ 6 \cdot \color{blue}{ \left( x-2 \right) }}{ \left( x+2 \right) \cdot \color{blue}{ \left( x-2 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 1 } }{ x^2-4 } + \frac{ \color{purple}{ 6x-12 } }{ x^2-4 }=\frac{ \color{purple}{ 6x-11 } }{ x^2-4 } \end{aligned} $$