Multiply $x-2$ by $ \dfrac{x-3}{2} $ to get $ \dfrac{x^2-5x+6}{2} $.

Step 1: Write $ x-2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} x-2 \cdot \frac{x-3}{2} & \xlongequal{\text{Step 1}} \frac{x-2}{\color{red}{1}} \cdot \frac{x-3}{2} \xlongequal{\text{Step 2}} \frac{ \left( x-2 \right) \cdot \left( x-3 \right) }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2-3x-2x+6 }{ 2 } = \frac{x^2-5x+6}{2} \end{aligned} $$

Add $ \dfrac{x^2-5x+6}{2} $ and $ 2 $ to get $ \dfrac{ \color{purple}{ x^2-5x+10 } }{ 2 }$.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{x^2-5x+6}{2} +2 & \xlongequal{\text{Step 1}} \frac{x^2-5x+6}{2} + \frac{2}{\color{red}{1}} = \frac{ x^2-5x+6 }{ 2 } + \frac{ 2 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2-5x+6 } }{ 2 } + \frac{ \color{purple}{ 4 } }{ 2 }=\frac{ \color{purple}{ x^2-5x+10 } }{ 2 } \end{aligned} $$