If ` alpha , beta ` are the roots of `ax^2+bx +c=0` then `(1+ alpha + alpha ^2)(1+ beta + beta ^2)` is

Options

A. `(1)/(a^(2))(a^(2)+b^(2)+c^(2)+ab+bc+ca)`
B. `(1)/(a^(2))(a^(2)+b^(2)+c^(2)-ab-bc-ca)`
C. `(1)/(a^(2))(a+b+c)`
D. `(1)/(a^(2))(a+b-c)^(2)`

Answer

To solve the problem, we need to find the value of the expression \( (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step-by-Step Solution:

1. **Identify the roots**:
Given the quadratic equation \( ax^2 + bx + c = 0 \), we know that:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \) (let's denote this as \( S \)).
- The product of the roots \( \alpha \beta = \frac{c}{a} \) (let's denote this as \( P \)).

2. **Rewrite the expression**:
We need to evaluate:
\[ A = (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) \]

3. **Expand the expression**:
Expanding \( A \):
\[ A = 1 \cdot (1 + \beta + \beta^2) + \alpha \cdot (1 + \beta + \beta^2) + \alpha^2 \cdot (1 + \beta + \beta^2) \]
This simplifies to:
\[ A = 1 + \beta + \beta^2 + \alpha + \alpha\beta + \alpha\beta^2 + \alpha^2 + \alpha^2\beta + \alpha^2\beta^2 \]

4. **Combine like terms**:
Grouping the terms:
\[ A = 1 + (\alpha + \beta) + (\alpha^2 + \beta^2) + \alpha\beta + (\alpha\beta^2 + \alpha^2\beta) + \alpha^2\beta^2 \]

5. **Use identities**:
We know:
- \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = S^2 - 2P \)
- \( \alpha\beta^2 + \alpha^2\beta = \alpha\beta(\alpha + \beta) = P \cdot S \)
- \( \alpha^2\beta^2 = (\alpha\beta)^2 = P^2 \)

6. **Substitute back**:
Substituting these into \( A \):
\[ A = 1 + S + (S^2 - 2P) + P + P \cdot S + P^2 \]
Simplifying this gives:
\[ A = 1 + S + S^2 - 2P + P + P \cdot S + P^2 \]
\[ A = 1 + S + S^2 - P + P \cdot S + P^2 \]

7. **Combine terms**:
Now, we can express \( A \) in terms of \( a, b, c \):
- Substitute \( S = -\frac{b}{a} \) and \( P = \frac{c}{a} \):
\[ A = 1 - \frac{b}{a} + \left(-\frac{b}{a}\right)^2 - \frac{c}{a} + \left(-\frac{b}{a}\right) \cdot \frac{c}{a} + \left(\frac{c}{a}\right)^2 \]

8. **Simplify**:
After simplification, we can find a common denominator \( a^2 \):
\[ A = \frac{a^2 - ab + b^2 - ac - bc + c^2}{a^2} \]

9. **Final Expression**:
Thus, we can conclude:
\[ A = \frac{1}{a^2}(a^2 + b^2 + c^2 - ab - ac - bc) \]

### Final Answer:
The value of \( (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) \) is:
\[ \frac{1}{a^2}(a^2 + b^2 + c^2 - ab - ac - bc) \]