The given equation in the image is $x^3 - x^2 - 10x - 8 = 0$, and the question is asking to find the value of $\alpha\beta + \beta\gamma + \gamma\alpha + \alpha\beta\gamma$, where $\alpha$, $\beta$, and $\gamma$ are the roots of the cubic equation.

Step 1: Understanding the problem

For a cubic equation of the form:

$x^3 + px^2 + qx + r = 0$

If $\alpha$, $\beta$, and $\gamma$ are the roots, we can use Vieta's formulas:

• $\alpha + \beta + \gamma = -p$ (sum of the roots),
• $\alpha\beta + \beta\gamma + \gamma\alpha = q$ (sum of the products of the roots taken two at a time),
• $\alpha\beta\gamma = -r$ (product of the roots).

Step 2: Identifying coefficients

Comparing the given equation $x^3 - x^2 - 10x - 8 = 0$ with the general form, we get:

• $p = -1$ (coefficient of $x^2$),
• $q = -10$ (coefficient of $x$),
• $r = -8$ (constant term).

Step 3: Using Vieta’s formulas

From Vieta’s formulas, we know:

• $\alpha + \beta + \gamma = -(-1) = 1$,
• $\alpha\beta + \beta\gamma + \gamma\alpha = -10$,
• $\alpha\beta\gamma = -(-8) = 8$.

Step 4: Calculating the required expression

The expression we need to find is $\alpha\beta + \beta\gamma + \gamma\alpha + \alpha\beta\gamma$.

Substituting the values from Vieta’s relations:

$\alpha\beta + \beta\gamma + \gamma\alpha + \alpha\beta\gamma = -10 + 8 = -2.$

Final Answer:

The value of $\alpha\beta + \beta\gamma + \gamma\alpha + \alpha\beta\gamma$ is $-2$.

So, the correct option is A: -2.

Let me know if you'd like further details or have any questions!

Here are five related questions:

1. How does Vieta's formula apply to quadratic equations?
2. Can Vieta's relations be used for equations of higher degree, and how?
3. What happens if one of the roots of the cubic equation is zero?
4. How would the expression change if we were given a quartic equation?
5. How can we factor a cubic polynomial given its roots?

Tip: Vieta's formulas are a powerful tool for quickly relating the coefficients of a polynomial to its roots without needing to solve the equation explicitly!