If ` alpha , beta , gamma ` are the roots of ` x^3 + 2x^2 + 3x +8=0` then ` ( alpha + beta ) ( beta + gamma) ( gamma + alpha )`=

Options

A. `-4`
B. `4`
C. `-2`
D. `2`

Answer

To solve the problem, we need to find the value of \( ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) \) given that \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 + 2x^2 + 3x + 8 = 0 \).

### Step 1: Identify the coefficients
The polynomial can be compared to the general form \( ax^3 + bx^2 + cx + d = 0 \). Here, we have:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 3 \)
- \( d = 8 \)

### Step 2: Use Vieta's formulas
According to Vieta's formulas for a cubic equation:
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{b}{a} = -\frac{2}{1} = -2 \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = \frac{3}{1} = 3 \)
- The product of the roots \( \alpha\beta\gamma = -\frac{d}{a} = -\frac{8}{1} = -8 \)

### Step 3: Express \( ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) \)
We can rewrite the expression:
\[ ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) = ( -\gamma )( -\alpha )( -\beta ) = -(\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha) + \alpha\beta\gamma \]

### Step 4: Substitute the known values
Now, substituting the values from Vieta's formulas:
\[ ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) = (-2)(3) + (-8) \]

### Step 5: Calculate the result
Calculating this gives:
\[ = -6 - 8 = -14 \]

### Step 6: Final expression
Thus, we have:
\[ ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) = -14 \]

However, we need to check our calculations as the options provided are different. Let's re-evaluate the expression \( ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) \) directly.

### Step 7: Re-evaluate the expression
Using the identity:
\[ ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) = (\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha) - \alpha\beta\gamma \]
Substituting the values:
\[ = (-2)(3) - (-8) = -6 + 8 = 2 \]

### Conclusion
Thus, the final answer is:
\[ ( \alpha + \beta )( \beta + \gamma )( \gamma + \alpha ) = 2 \]