The problem states that if $\alpha$ and $\beta$ are the roots of the quadratic equation:

$2x^2 - 3x - 6 = 0$

we need to find the equation whose roots are $\alpha^2 + 2$ and $\beta^2 + 2$.

Step 1: Use Vieta's Formulas

From Vieta’s formulas for a quadratic equation $ax^2 + bx + c = 0$, we know the following relationships:

• Sum of the roots $\alpha + \beta = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = \frac{-(-3)}{2} = \frac{3}{2}$
• Product of the roots $\alpha \beta = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{-6}{2} = -3$

Step 2: Find New Roots

We now need the roots of the new equation where the roots are $\alpha^2 + 2$ and $\beta^2 + 2$.

• The sum of the new roots is: $(\alpha^2 + 2) + (\beta^2 + 2) = \alpha^2 + \beta^2 + 4$

Using the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$, we get: $\alpha^2 + \beta^2 = \left( \frac{3}{2} \right)^2 - 2(-3) = \frac{9}{4} + 6 = \frac{33}{4}$

Thus, the sum of the new roots is: $\frac{33}{4} + 4 = \frac{49}{4}$

• The product of the new roots is: $(\alpha^2 + 2)(\beta^2 + 2) = \alpha^2 \beta^2 + 2\alpha^2 + 2\beta^2 + 4$

Since $\alpha^2 \beta^2 = (\alpha \beta)^2 = (-3)^2 = 9$, and $\alpha^2 + \beta^2 = \frac{33}{4}$, the product becomes: $9 + 2 \times \frac{33}{4} + 4 = 9 + \frac{66}{4} + 4 = 13 + \frac{66}{4} = \frac{118}{4}$

Step 3: Form the Equation

The quadratic equation whose roots have a sum of $\frac{49}{4}$ and a product of $\frac{118}{4}$ is given by:

$4x^2 - 49x + 118 = 0$

This matches option 4 in the provided choices.

Would you like more details on any step? Here are some follow-up questions to explore this topic further:

1. How does Vieta’s formula help in solving quadratic problems?
2. Can you derive the sum and product of new roots for transformed equations in general?
3. What is the significance of the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$?
4. How would the process change if we were solving for cubics or higher degree polynomials?
5. What techniques can be used to verify the solution for transformed roots?

Tip: When working with quadratic equations, always try to simplify the process using identities like Vieta’s relations to avoid solving the equation from scratch.