Add $ \dfrac{4}{x^2-7x+6} $ and $ \dfrac{x}{x-6} $ to get $ \dfrac{ \color{purple}{ x^2-x+4 } }{ x^2-7x+6 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x-1}$.

$$ \begin{aligned} \frac{4}{x^2-7x+6} + \frac{x}{x-6} & = \frac{ 4 }{ x^2-7x+6 } + \frac{ x \cdot \color{blue}{ \left( x-1 \right) }}{ \left( x-6 \right) \cdot \color{blue}{ \left( x-1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 4 } }{ x^2-7x+6 } + \frac{ \color{purple}{ x^2-x } }{ x^2-7x+6 }=\frac{ \color{purple}{ x^2-x+4 } }{ x^2-7x+6 } \end{aligned} $$