Add $ \dfrac{6}{a+1} $ and $ \dfrac{5}{a-3} $ to get $ \dfrac{ \color{purple}{ 11a-13 } }{ a^2-2a-3 }$.

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}{ a-3 }$ and the second by $\color{blue}{ a+1 }$.

$$ \begin{aligned} \frac{6}{a+1} + \frac{5}{a-3} & = \frac{ 6 \cdot \color{blue}{ \left( a-3 \right) }}{ \left( a+1 \right) \cdot \color{blue}{ \left( a-3 \right) }} + \frac{ 5 \cdot \color{blue}{ \left( a+1 \right) }}{ \left( a-3 \right) \cdot \color{blue}{ \left( a+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 6a-18 } }{ a^2-3a+a-3 } + \frac{ \color{purple}{ 5a+5 } }{ a^2-3a+a-3 }=\frac{ \color{purple}{ 11a-13 } }{ a^2-2a-3 } \end{aligned} $$