Find $ \left(x+2\right)^2 $ using formula.

$$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 2 }$.

$$ \begin{aligned}\left(x+2\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 2 + \color{red}{2^2} = x^2+4x+4\end{aligned} $$

Add $ \dfrac{2}{x+2} $ and $ \dfrac{3}{x^2+4x+4} $ to get $ \dfrac{ \color{purple}{ 2x+7 } }{ x^2+4x+4 }$.

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

$$ \begin{aligned} \frac{2}{x+2} + \frac{3}{x^2+4x+4} & = \frac{ 2 \cdot \color{blue}{ \left( x+2 \right) }}{ \left( x+2 \right) \cdot \color{blue}{ \left( x+2 \right) }} + \frac{ 3 }{ x^2+4x+4 } = \\[1ex] &=\frac{ \color{purple}{ 2x+4 } }{ x^2+2x+2x+4 } + \frac{ \color{purple}{ 3 } }{ x^2+2x+2x+4 }=\frac{ \color{purple}{ 2x+7 } }{ x^2+4x+4 } \end{aligned} $$