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

Answer

To solve the problem, we need to find the expression \((\beta + \gamma - 3\alpha)(\gamma + \alpha - 3\beta)(\alpha + \beta - 3\gamma)\) given that \(\alpha\), \(\beta\), and \(\gamma\) are the roots of the polynomial \(x^3 + px^2 + qx + r = 0\).

### Step 1: Use Vieta's Formulas
From Vieta's formulas, we know:
- The sum of the roots: \(\alpha + \beta + \gamma = -p\)
- The sum of the products of the roots taken two at a time: \(\alpha\beta + \beta\gamma + \gamma\alpha = q\)
- The product of the roots: \(\alpha\beta\gamma = -r\)

### Step 2: Rewrite the Expression
We can rewrite the expression we need to evaluate:
\[ (\beta + \gamma - 3\alpha)(\gamma + \alpha - 3\beta)(\alpha + \beta - 3\gamma) \]
Using the identity \(\beta + \gamma = -p - \alpha\), we can express each term in the product:
1. \(\beta + \gamma - 3\alpha = -p - \alpha - 3\alpha = -p - 4\alpha\)
2. \(\gamma + \alpha - 3\beta = -p - \beta - 3\beta = -p - 4\beta\)
3. \(\alpha + \beta - 3\gamma = -p - \gamma - 3\gamma = -p - 4\gamma\)

### Step 3: Substitute Back into the Expression
Now substituting these back into the expression, we have:
\[ (-p - 4\alpha)(-p - 4\beta)(-p - 4\gamma) \]

### Step 4: Factor Out the Negative Sign
Factoring out the negative sign, we get:
\[ -(p + 4\alpha)(p + 4\beta)(p + 4\gamma) \]

### Step 5: Expand the Product
Now we expand the product:
\[ = -(p^3 + 4p^2(\alpha + \beta + \gamma) + 16p(\alpha\beta + \beta\gamma + \gamma\alpha) + 64\alpha\beta\gamma) \]

### Step 6: Substitute Values from Vieta's Formulas
Substituting the values from Vieta's formulas:
- \(\alpha + \beta + \gamma = -p\)
- \(\alpha\beta + \beta\gamma + \gamma\alpha = q\)
- \(\alpha\beta\gamma = -r\)

We have:
\[ = -(p^3 + 4p^2(-p) + 16pq + 64(-r)) \]
\[ = -(p^3 - 4p^3 + 16pq - 64r) \]
\[ = -(-3p^3 + 16pq - 64r) \]
\[ = 3p^3 - 16pq + 64r \]

### Final Answer
Thus, the required expression evaluates to:
\[ \boxed{3p^3 - 16pq + 64r} \]