If `alpha and beta` are the roots the equations `ax^(2) +bx + c = 0`, where `a!=0` then `(a alpha + b) (a beta +b)` is equal to :

Options

A. ab
B. bc
C. ca
D. abc

Answer

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

### Step-by-step Solution:

1. **Identify the expression to evaluate**:
We need to evaluate the expression \((a \alpha + b)(a \beta + b)\).

2. **Expand the expression**:
Using the distributive property (FOIL method), we can expand the expression:
\[ (a \alpha + b)(a \beta + b) = a^2 \alpha \beta + ab \alpha + ab \beta + b^2 \]
This simplifies to:
\[ = a^2 \alpha \beta + ab(\alpha + \beta) + b^2 \]

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

4. **Substitute the values from Vieta's**:
Substitute \(\alpha + \beta\) and \(\alpha \beta\) into the expanded expression:
\[ = a^2 \left(\frac{c}{a}\right) + ab\left(-\frac{b}{a}\right) + b^2 \]
This simplifies to:
\[ = ac - \frac{b^2}{a} + b^2 \]

5. **Combine like terms**:
To combine the terms, we can express \(b^2\) with a common denominator:
\[ = ac - \frac{b^2}{a} + \frac{b^2 a}{a} = ac + \frac{b^2 a - b^2}{a} \]
This simplifies to:
\[ = ac + \frac{b^2(a - 1)}{a} \]

6. **Final expression**:
Therefore, the final expression for \((a \alpha + b)(a \beta + b)\) is:
\[ = ac + \frac{b^2(a - 1)}{a} \]

### Conclusion:
Thus, the value of \((a \alpha + b)(a \beta + b)\) is \(ac + \frac{b^2(a - 1)}{a}\).