Add $ \dfrac{p}{p+2} $ and $ \dfrac{2}{p^2+5p+6} $ to get $ \dfrac{ \color{purple}{ p^2+3p+2 } }{ p^2+5p+6 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ p+3 }$.

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

Simplify $ \dfrac{p^2+3p+2}{p^2+5p+6} $ to $ \dfrac{p+1}{p+3} $.

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

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