Add $ \dfrac{x-5}{x^2+4x+3} $ and $ \dfrac{2}{5} $ to get $ \dfrac{ \color{purple}{ 2x^2+13x-19 } }{ 5x^2+20x+15 }$.

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}{ 5 }$ and the second by $\color{blue}{ x^2+4x+3 }$.

$$ \begin{aligned} \frac{x-5}{x^2+4x+3} + \frac{2}{5} & = \frac{ \left( x-5 \right) \cdot \color{blue}{ 5 }}{ \left( x^2+4x+3 \right) \cdot \color{blue}{ 5 }} + \frac{ 2 \cdot \color{blue}{ \left( x^2+4x+3 \right) }}{ 5 \cdot \color{blue}{ \left( x^2+4x+3 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 5x-25 } }{ 5x^2+20x+15 } + \frac{ \color{purple}{ 2x^2+8x+6 } }{ 5x^2+20x+15 } = \\[1ex] &=\frac{ \color{purple}{ 2x^2+13x-19 } }{ 5x^2+20x+15 } \end{aligned} $$