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

To add 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 }$ and the second by $\color{blue}{ x+2 }$.

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