Subtract $ \dfrac{2x}{x+4} $ from $ \dfrac{x+5}{x^2-16} $ to get $ \dfrac{ \color{purple}{ -2x^2+9x+5 } }{ x^2-16 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x-4}$.

$$ \begin{aligned} \frac{x+5}{x^2-16} - \frac{2x}{x+4} & = \frac{ x+5 }{ x^2-16 } - \frac{ 2x \cdot \color{blue}{ \left( x-4 \right) }}{ \left( x+4 \right) \cdot \color{blue}{ \left( x-4 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x+5 } }{ x^2-16 } - \frac{ \color{purple}{ 2x^2-8x } }{ x^2-16 }=\frac{ \color{purple}{ x+5 - \left( 2x^2-8x \right) } }{ x^2-16 } = \\[1ex] &=\frac{ \color{purple}{ -2x^2+9x+5 } }{ x^2-16 } \end{aligned} $$