`(a + b - 2c) ^(3) + (b + c - 2a) ^(3) + (c + a - 2b) ^(3)` is equal to :

Options

A. `(a + b - 2c) (b + c - 2a) (c = a - 2b)`
B. `2 (a + b - 2c) (b + c - 2a) (c + a - 2b)`
C. `3(a + b - 2c) (b + c - 2a) (c + a - 2b)`
D. `-3a (a + b - 2c) (b + c - 2a) (c + a - 2b)`

Answer

To solve the expression \((a + b - 2c)^3 + (b + c - 2a)^3 + (c + a - 2b)^3\), we can use a well-known algebraic identity.

### Step-by-Step Solution:

1. **Identify the terms**:
Let:
- \(x = a + b - 2c\)
- \(y = b + c - 2a\)
- \(z = c + a - 2b\)

We need to evaluate \(x^3 + y^3 + z^3\).

2. **Check if \(x + y + z = 0\)**:
Calculate \(x + y + z\):
\[ x + y + z = (a + b - 2c) + (b + c - 2a) + (c + a - 2b) \]
Simplifying this:
\[ = a + b - 2c + b + c - 2a + c + a - 2b \]
Combine like terms:
\[ = (a - 2a + a) + (b - 2b + b) + (-2c + c + c) = 0 \]

Since \(x + y + z = 0\), we can use the identity for cubes.

3. **Apply the identity**:
The identity states that if \(x + y + z = 0\), then:
\[ x^3 + y^3 + z^3 = 3xyz \]

4. **Calculate \(xyz\)**:
Now we need to find \(xyz\):
\[ xyz = (a + b - 2c)(b + c - 2a)(c + a - 2b) \]

5. **Substituting back**:
Therefore, we have:
\[ (a + b - 2c)^3 + (b + c - 2a)^3 + (c + a - 2b)^3 = 3xyz \]

6. **Final Expression**:
The final expression is:
\[ 3(a + b - 2c)(b + c - 2a)(c + a - 2b) \]

### Conclusion:
Thus, the expression \((a + b - 2c)^3 + (b + c - 2a)^3 + (c + a - 2b)^3\) simplifies to \(3(a + b - 2c)(b + c - 2a)(c + a - 2b)\).