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

Answer

To find the value of \((\alpha + \beta)(\beta + \gamma)(\gamma + \alpha)\) given that \(\alpha, \beta, \gamma\) are the roots of the polynomial equation \(x^3 + px^2 + qx + r = 0\), we can follow these steps:

### Step 1: Identify the roots and coefficients
The roots of the polynomial are \(\alpha, \beta, \gamma\) and the polynomial can be expressed in the standard form \(ax^3 + bx^2 + cx + d = 0\). Here, we have:
- \(a = 1\)
- \(b = p\)
- \(c = q\)
- \(d = r\)

### Step 2: Use Vieta's formulas
From Vieta's formulas, we know:
1. \(\alpha + \beta + \gamma = -\frac{b}{a} = -p\)
2. \(\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} = q\)
3. \(\alpha \beta \gamma = -\frac{d}{a} = -r\)

### Step 3: Expand the expression
We need to expand \((\alpha + \beta)(\beta + \gamma)(\gamma + \alpha)\):
\[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) = (\alpha + \beta + \gamma)(\alpha \beta + \beta \gamma + \gamma \alpha) - \alpha \beta \gamma \]
Substituting the values from Vieta's formulas:
- \(\alpha + \beta + \gamma = -p\)
- \(\alpha \beta + \beta \gamma + \gamma \alpha = q\)
- \(\alpha \beta \gamma = -r\)

### Step 4: Substitute the values
Now substituting these values into the expanded expression:
\[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) = (-p)(q) - (-r) \]
This simplifies to:
\[ -pq + r \]

### Step 5: Final result
Thus, the value of \((\alpha + \beta)(\beta + \gamma)(\gamma + \alpha)\) is:
\[ r - pq \]

### Conclusion
The final answer is:
\[ \boxed{r - pq} \]