Add $ \dfrac{x-3}{x+5} $ and $ \dfrac{x^2+3x+2}{x-3} $ to get $ \dfrac{ \color{purple}{ x^3+9x^2+11x+19 } }{ x^2+2x-15 }$.

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-3 }$ and the second by $\color{blue}{ x+5 }$.

$$ \begin{aligned} \frac{x-3}{x+5} + \frac{x^2+3x+2}{x-3} & = \frac{ \left( x-3 \right) \cdot \color{blue}{ \left( x-3 \right) }}{ \left( x+5 \right) \cdot \color{blue}{ \left( x-3 \right) }} + \frac{ \left( x^2+3x+2 \right) \cdot \color{blue}{ \left( x+5 \right) }}{ \left( x-3 \right) \cdot \color{blue}{ \left( x+5 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x^2-3x-3x+9 } }{ x^2-3x+5x-15 } + \frac{ \color{purple}{ x^3+5x^2+3x^2+15x+2x+10 } }{ x^2-3x+5x-15 } = \\[1ex] &=\frac{ \color{purple}{ x^3+9x^2+11x+19 } }{ x^2+2x-15 } \end{aligned} $$