If `alpha, beta` are the roots of the quadratic equation `x^2 + bx - c = 0`, the equation whose roots are `b` and `c`, is a. `x^(2)+alpha x- beta=0` b. `x^(2)-[(alpha +beta)+alpha beta]x-alpha beta( alpha+beta)=0` c. `x^(2)+[(alpha + beta)+alpha beta]x+alpha beta(alpha + beta)=0` d. `x^(2)+[(alpha +beta)+alpha beta)]x -alpha beta(alpha +beta)=0`

Options

A. `x^(2)+alpha x- beta-0`
B. `x^(2)-[(alpha +beta)+alpha beta]x-alpha beta( alpha+beta)=0`
C. `x^(2)+[(alpha + beta)+alpha beta]x+alpha beta(alpha + beta)=0`
D. `x^(2)+[(alpha +beta)+alpha beta)]x -alpha beta(alpha +beta)=0`

Answer

To solve the problem, we need to find the quadratic equation whose roots are \( b \) and \( c \), given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 + bx - c = 0 \).

### Step-by-step Solution:

1. **Identify the given quadratic equation**:
The equation is given as:
\[ x^2 + bx - c = 0 \]
Here, the roots are \( \alpha \) and \( \beta \).

2. **Use Vieta's Formulas**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -b \) (Equation 1)
- The product of the roots \( \alpha \beta = -c \) (Equation 2)

3. **Express \( b \) and \( c \)**:
From Equation 1, we can express \( b \) as:
\[ b = -(\alpha + \beta) \]
From Equation 2, we can express \( c \) as:
\[ c = -\alpha \beta \]

4. **Find the sum and product of the new roots \( b \) and \( c \)**:
- The sum of the roots \( b + c \):
\[ b + c = -(\alpha + \beta) - \alpha \beta \]
- The product of the roots \( b \cdot c \):
\[ b \cdot c = (-(\alpha + \beta)) \cdot (-\alpha \beta) = \alpha \beta (\alpha + \beta) \]

5. **Form the new quadratic equation**:
The general form of a quadratic equation with roots \( r_1 \) and \( r_2 \) is:
\[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \]
Substituting \( r_1 = b \) and \( r_2 = c \):
\[ x^2 - (b + c)x + (b \cdot c) = 0 \]
Substituting the values we found:
\[ x^2 + \left( -(\alpha + \beta) - \alpha \beta \right)x + \left( \alpha \beta (\alpha + \beta) \right) = 0 \]
This simplifies to:
\[ x^2 + \left( \alpha + \beta + \alpha \beta \right)x + \alpha \beta (\alpha + \beta) = 0 \]

6. **Final equation**:
The final quadratic equation is:
\[ x^2 + \left( \alpha + \beta + \alpha \beta \right)x + \alpha \beta (\alpha + \beta) = 0 \]

### Conclusion:
The correct option is:
**c. \( x^2 + [(\alpha + \beta) + \alpha \beta]x + \alpha \beta (\alpha + \beta) = 0 \)**