Add $ \dfrac{5x}{x^2-100} $ and $ \dfrac{50}{x+10} $ to get $ \dfrac{ \color{purple}{ 55x-500 } }{ x^2-100 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x-10}$.

$$ \begin{aligned} \frac{5x}{x^2-100} + \frac{50}{x+10} & = \frac{ 5x }{ x^2-100 } + \frac{ 50 \cdot \color{blue}{ \left( x-10 \right) }}{ \left( x+10 \right) \cdot \color{blue}{ \left( x-10 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 5x } }{ x^2-100 } + \frac{ \color{purple}{ 50x-500 } }{ x^2-100 }=\frac{ \color{purple}{ 55x-500 } }{ x^2-100 } \end{aligned} $$