Add $ \dfrac{6}{p+2} $ and $ \dfrac{8}{p^2-4} $ to get $ \dfrac{ \color{purple}{ 6p-4 } }{ p^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 $\color{blue}{ p-2 }$.

$$ \begin{aligned} \frac{6}{p+2} + \frac{8}{p^2-4} & = \frac{ 6 \cdot \color{blue}{ \left( p-2 \right) }}{ \left( p+2 \right) \cdot \color{blue}{ \left( p-2 \right) }} + \frac{ 8 }{ p^2-4 } = \\[1ex] &=\frac{ \color{purple}{ 6p-12 } }{ p^2 -\cancel{2p}+ \cancel{2p}-4 } + \frac{ \color{purple}{ 8 } }{ p^2 -\cancel{2p}+ \cancel{2p}-4 } = \\[1ex] &=\frac{ \color{purple}{ 6p-4 } }{ p^2-4 } \end{aligned} $$

Divide $ \dfrac{2}{p-2} $ by $ \dfrac{6p-4}{p^2-4} $ to get $ \dfrac{ 2p+4 }{ 6p-4 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{2}{p-2} }{ \frac{\color{blue}{6p-4}}{\color{blue}{p^2-4}} } & \xlongequal{\text{Step 1}} \frac{2}{p-2} \cdot \frac{\color{blue}{p^2-4}}{\color{blue}{6p-4}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 2 }{ 1 \cdot \color{red}{ \left( p-2 \right) } } \cdot \frac{ \left( p+2 \right) \cdot \color{red}{ \left( p-2 \right) } }{ 6p-4 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2 }{ 1 } \cdot \frac{ p+2 }{ 6p-4 } \xlongequal{\text{Step 4}} \frac{ 2 \cdot \left( p+2 \right) }{ 1 \cdot \left( 6p-4 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ 2p+4 }{ 6p-4 } \end{aligned} $$