Add $ \dfrac{5}{x-7} $ and $ \dfrac{3}{x-1} $ to get $ \dfrac{ \color{purple}{ 8x-26 } }{ x^2-8x+7 }$.

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-1 }$ and the second by $\color{blue}{ x-7 }$.

$$ \begin{aligned} \frac{5}{x-7} + \frac{3}{x-1} & = \frac{ 5 \cdot \color{blue}{ \left( x-1 \right) }}{ \left( x-7 \right) \cdot \color{blue}{ \left( x-1 \right) }} + \frac{ 3 \cdot \color{blue}{ \left( x-7 \right) }}{ \left( x-1 \right) \cdot \color{blue}{ \left( x-7 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 5x-5 } }{ x^2-x-7x+7 } + \frac{ \color{purple}{ 3x-21 } }{ x^2-x-7x+7 }=\frac{ \color{purple}{ 8x-26 } }{ x^2-8x+7 } \end{aligned} $$