If root of the equation `x^(2) + ax + b = 0 are alpha , beta` , then the roots of `x^(2) + (2 alpha + a) x + alpha^(2) + a alpha + b = 0` are

Options

A. `1, beta- alpha`
B. `0, alpha - beta`
C. `0, beta - alpha`
D. 0,1

Answer

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

### Step-by-Step Solution:

1. **Identify the given quadratic equation**:
The original equation is:
\[ x^2 + ax + b = 0 \]
with roots \(\alpha\) and \(\beta\).

2. **Use Vieta's formulas**:
From Vieta's formulas, we know:
- The sum of the roots: \(\alpha + \beta = -a\)
- The product of the roots: \(\alpha \beta = b\)

3. **Consider the new quadratic equation**:
The new quadratic equation we need to analyze is:
\[ x^2 + (2\alpha + a)x + (\alpha^2 + a\alpha + b) = 0 \]

4. **Substitute \(\alpha\) and \(\beta\)**:
We know from the original equation that:
\[ \alpha^2 + a\alpha + b = 0 \]
Therefore, we can replace \(\alpha^2 + a\alpha + b\) in the new equation:
\[ x^2 + (2\alpha + a)x + 0 = 0 \]

5. **Simplify the equation**:
This simplifies to:
\[ x^2 + (2\alpha + a)x = 0 \]

6. **Factor the equation**:
We can factor out \(x\):
\[ x(x + (2\alpha + a)) = 0 \]

7. **Find the roots**:
Setting each factor to zero gives us the roots:
\[ x = 0 \quad \text{or} \quad x + (2\alpha + a) = 0 \]
Thus, the second root is:
\[ x = - (2\alpha + a) \]

### Final Roots:
The roots of the quadratic equation \(x^2 + (2\alpha + a)x + (\alpha^2 + a\alpha + b) = 0\) are:
\[ x = 0 \quad \text{and} \quad x = - (2\alpha + a) \]