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

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-2 }$ and the second by $\color{blue}{ x-1 }$.

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