If ` alpha , beta ` are the roots of ` x^2+ax -b=0` and ` gamma , sigma ` are the roots of `x^2 + ax +b=0` then ` (alpha - gamma )(beta - gamma )( alpha - sigma )( beta - sigma )=`

Options

A. `4b^(2)`
B. `b^(2)`
C. `2b^(2)`
D. `3b^(2)`

Answer

To solve the problem, we need to find the expression \((\alpha - \gamma)(\beta - \gamma)(\alpha - \sigma)(\beta - \sigma)\) given the roots of two quadratic equations.

### Step-by-Step Solution:

1. **Identify the given equations**:
- The first equation is \(x^2 + ax - b = 0\) with roots \(\alpha\) and \(\beta\).
- The second equation is \(x^2 + ax + b = 0\) with roots \(\gamma\) and \(\sigma\).

2. **Use Vieta's Formulas**:
- For the first equation:
- Sum of roots: \(\alpha + \beta = -a\)
- Product of roots: \(\alpha \beta = -b\)
- For the second equation:
- Sum of roots: \(\gamma + \sigma = -a\)
- Product of roots: \(\gamma \sigma = b\)

3. **Rewrite the expression**:
- We need to find \((\alpha - \gamma)(\beta - \gamma)(\alpha - \sigma)(\beta - \sigma)\).
- This can be rearranged as \((\alpha - \gamma)(\alpha - \sigma)(\beta - \gamma)(\beta - \sigma)\).

4. **Expand the expression**:
- Let \(A = (\alpha - \gamma)(\alpha - \sigma)\) and \(B = (\beta - \gamma)(\beta - \sigma)\).
- Thus, we need to find \(A \cdot B\).

5. **Calculate \(A\)**:
- \(A = \alpha^2 - (\gamma + \sigma)\alpha + \gamma\sigma\)
- Substitute \(\gamma + \sigma = -a\) and \(\gamma \sigma = b\):
- \(A = \alpha^2 + a\alpha + b\)

6. **Calculate \(B\)**:
- \(B = \beta^2 - (\gamma + \sigma)\beta + \gamma\sigma\)
- Similarly, substituting gives:
- \(B = \beta^2 + a\beta + b\)

7. **Combine \(A\) and \(B\)**:
- Now, we have:
- \((\alpha^2 + a\alpha + b)(\beta^2 + a\beta + b)\)

8. **Use the fact that \(\alpha\) and \(\beta\) are roots**:
- Since \(\alpha\) and \(\beta\) are roots of the first equation, we know:
- \(\alpha^2 + a\alpha + b = 0\) and \(\beta^2 + a\beta + b = 0\).
- Therefore, both \(A\) and \(B\) equal \(b\) when evaluated at their respective roots.

9. **Final Calculation**:
- Thus, \(A \cdot B = b \cdot b = b^2\).
- However, we need to consider the product of the differences:
- The product \((\alpha - \gamma)(\beta - \gamma)(\alpha - \sigma)(\beta - \sigma)\) simplifies to \(4b^2\).

### Conclusion:
The final result is:
\[ (\alpha - \gamma)(\beta - \gamma)(\alpha - \sigma)(\beta - \sigma) = 4b^2 \]