Question:

Step 1: Extract the relationships from the first equation. From the equation $x^2 - mx + 2 = 0$ with roots $\alpha$ and $\beta$, we apply Vieta's formulas: $$\alpha + \beta = m$$ $$\alpha\beta = 2$$ Step 2: Identify the roots of the second equation. The roots of the equation $x^2 - px + q = 0$ are given as: $$\alpha + \frac{1}{\beta} \quad \text{and} \quad \beta + \frac{1}{\alpha}$$ Step 3: Find the product of the roots of the second equation. By Vieta's formulas, the product of the roots equals $q$: $$q = \left(\alpha + \frac{1}{\beta}\right)\left(\beta + \frac{1}{\alpha}\right)$$ Step 4: Expand the product. Expanding the expression: $$q = \alpha\beta + \alpha \cdot \frac{1}{\alpha} + \frac{1}{\beta} \cdot \beta + \frac{1}{\beta} \cdot \frac{1}{\alpha}$$ $$q = \alpha\beta + 1 + 1 + \frac{1}{\alpha\beta}$$ Step 5: Substitute the known values. Using $\alpha\beta = 2$ from Step 1: $$q = 2 + 1 + 1 + \frac{1}{2}$$ $$q = 4 + \frac{1}{2} = \frac{9}{2}$$ Step 6: Calculate $2q$. $$2q = 2 \cdot \frac{9}{2} = 9$$ **Final Answer:** The value of $2q$ equals $\boxed{9}$, which corresponds to **Option 4**.