Add $ \dfrac{3x-1}{x^2+3x+2} $ and $ \dfrac{4}{x+1} $ to get $ \dfrac{ \color{purple}{ 7x+7 } }{ x^2+3x+2 }$.

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

$$ \begin{aligned} \frac{3x-1}{x^2+3x+2} + \frac{4}{x+1} & = \frac{ 3x-1 }{ x^2+3x+2 } + \frac{ 4 \cdot \color{blue}{ \left( x+2 \right) }}{ \left( x+1 \right) \cdot \color{blue}{ \left( x+2 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3x-1 } }{ x^2+3x+2 } + \frac{ \color{purple}{ 4x+8 } }{ x^2+3x+2 }=\frac{ \color{purple}{ 7x+7 } }{ x^2+3x+2 } \end{aligned} $$

Simplify $ \dfrac{7x+7}{x^2+3x+2} $ to $ \dfrac{7}{x+2} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+1}$.

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