Add $ \dfrac{x}{x-1} $ and $ \dfrac{4}{x+3} $ to get $ \dfrac{ \color{purple}{ x^2+7x-4 } }{ x^2+2x-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}{ x+3 }$ and the second by $\color{blue}{ x-1 }$.

$$ \begin{aligned} \frac{x}{x-1} + \frac{4}{x+3} & = \frac{ x \cdot \color{blue}{ \left( x+3 \right) }}{ \left( x-1 \right) \cdot \color{blue}{ \left( x+3 \right) }} + \frac{ 4 \cdot \color{blue}{ \left( x-1 \right) }}{ \left( x+3 \right) \cdot \color{blue}{ \left( x-1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x^2+3x } }{ x^2+3x-x-3 } + \frac{ \color{purple}{ 4x-4 } }{ x^2+3x-x-3 }=\frac{ \color{purple}{ x^2+7x-4 } }{ x^2+2x-3 } \end{aligned} $$