Add $ \dfrac{3}{x+7} $ and $ \dfrac{4}{x-8} $ to get $ \dfrac{ \color{purple}{ 7x+4 } }{ x^2-x-56 }$.

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-8 }$ and the second by $\color{blue}{ x+7 }$.

$$ \begin{aligned} \frac{3}{x+7} + \frac{4}{x-8} & = \frac{ 3 \cdot \color{blue}{ \left( x-8 \right) }}{ \left( x+7 \right) \cdot \color{blue}{ \left( x-8 \right) }} + \frac{ 4 \cdot \color{blue}{ \left( x+7 \right) }}{ \left( x-8 \right) \cdot \color{blue}{ \left( x+7 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3x-24 } }{ x^2-8x+7x-56 } + \frac{ \color{purple}{ 4x+28 } }{ x^2-8x+7x-56 }=\frac{ \color{purple}{ 7x+4 } }{ x^2-x-56 } \end{aligned} $$