Add $ \dfrac{x}{x^2-16} $ and $ \dfrac{4}{x^2+6x+8} $ to get $ \dfrac{ \color{purple}{ x^2+6x-16 } }{ x^3+2x^2-16x-32 }$.

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+2 }$ and the second by $\color{blue}{ x-4 }$.

$$ \begin{aligned} \frac{x}{x^2-16} + \frac{4}{x^2+6x+8} & = \frac{ x \cdot \color{blue}{ \left( x+2 \right) }}{ \left( x^2-16 \right) \cdot \color{blue}{ \left( x+2 \right) }} + \frac{ 4 \cdot \color{blue}{ \left( x-4 \right) }}{ \left( x^2+6x+8 \right) \cdot \color{blue}{ \left( x-4 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x^2+2x } }{ x^3+2x^2-16x-32 } + \frac{ \color{purple}{ 4x-16 } }{ x^3+2x^2-16x-32 } = \\[1ex] &=\frac{ \color{purple}{ x^2+6x-16 } }{ x^3+2x^2-16x-32 } \end{aligned} $$