Multiply each term of $ \left( \color{blue}{x+2}\right) $ by each term in $ \left( x+2\right) $.

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

Combine like terms:

$$ x^2+ \color{blue}{2x} + \color{blue}{2x} +4 = x^2+ \color{blue}{4x} +4 $$

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 $\color{blue}{ x+2 }$.

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