Add $ \dfrac{4}{x+5} $ and $ \dfrac{6x-7}{x^2+6x+5} $ to get $ \dfrac{ \color{purple}{ 10x-3 } }{ x^2+6x+5 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ x+1 }$.

$$ \begin{aligned} \frac{4}{x+5} + \frac{6x-7}{x^2+6x+5} & = \frac{ 4 \cdot \color{blue}{ \left( x+1 \right) }}{ \left( x+5 \right) \cdot \color{blue}{ \left( x+1 \right) }} + \frac{ 6x-7 }{ x^2+6x+5 } = \\[1ex] &=\frac{ \color{purple}{ 4x+4 } }{ x^2+x+5x+5 } + \frac{ \color{purple}{ 6x-7 } }{ x^2+x+5x+5 }=\frac{ \color{purple}{ 10x-3 } }{ x^2+6x+5 } \end{aligned} $$