One of the factor of `(a+2b)^(3)+(2a-c)^(3)-(a+2c)^(3)+3(a+2b)(2a-c)(a+2c)` is

Options

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

Answer

To find one of the factors of the expression \((a + 2b)^3 + (2a - c)^3 - (a + 2c)^3 + 3(a + 2b)(2a - c)(a + 2c)\), we can use the identity for the sum of cubes.

### Step-by-Step Solution:

1. **Identify the Variables**:
We will let:
- \(x = a + 2b\)
- \(y = 2a - c\)
- \(z = -(a + 2c)\)

This allows us to rewrite the expression in terms of \(x\), \(y\), and \(z\).

2. **Apply the Identity**:
The identity we will use is:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) \]
Here, we can see that our expression can be rearranged to fit this identity.

3. **Rewrite the Expression**:
The expression can be rewritten as:
\[ (a + 2b)^3 + (2a - c)^3 + (-(a + 2c))^3 + 3(a + 2b)(2a - c)(-(a + 2c)) \]

4. **Calculate \(x + y + z\)**:
Now we calculate:
\[ x + y + z = (a + 2b) + (2a - c) - (a + 2c) = 2a + 2b - 3c \]

5. **Calculate \(x^2 + y^2 + z^2 - xy - xz - yz\)**:
We need to compute:
- \(x^2 = (a + 2b)^2\)
- \(y^2 = (2a - c)^2\)
- \(z^2 = (-(a + 2c))^2\)
- \(xy = (a + 2b)(2a - c)\)
- \(xz = (a + 2b)(-(a + 2c))\)
- \(yz = (2a - c)(-(a + 2c))\)

However, since we only need one factor, we can stop here.

6. **Conclusion**:
From our calculations, we see that one of the factors of the original expression is:
\[ 2a + 2b - 3c \]

### Final Answer:
Thus, one of the factors of the expression is:
\[ \boxed{2a + 2b - 3c} \]