Factorise
` (2a+ b)^(3) - (a+ 2b)^(3)`

Answer

To factorize the expression \( (2a + b)^3 - (a + 2b)^3 \), we can use the identity for the difference of cubes, which states that:

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

where \( x = 2a + b \) and \( y = a + 2b \).

### Step 1: Identify \( x \) and \( y \)
Let:
- \( x = 2a + b \)
- \( y = a + 2b \)

### Step 2: Calculate \( x - y \)
Now, we calculate \( x - y \):
\[ x - y = (2a + b) - (a + 2b) = 2a + b - a - 2b = a - b \]

### Step 3: Calculate \( x^2 + xy + y^2 \)
Next, we need to calculate \( x^2 + xy + y^2 \).

1. **Calculate \( x^2 \)**:
\[ x^2 = (2a + b)^2 = 4a^2 + 4ab + b^2 \]

2. **Calculate \( y^2 \)**:
\[ y^2 = (a + 2b)^2 = a^2 + 4ab + 4b^2 \]

3. **Calculate \( xy \)**:
\[ xy = (2a + b)(a + 2b) = 2a^2 + 4ab + ab + 2b^2 = 2a^2 + 5ab + 2b^2 \]

4. **Combine \( x^2 + xy + y^2 \)**:
\[ x^2 + xy + y^2 = (4a^2 + 4ab + b^2) + (2a^2 + 5ab + 2b^2) + (a^2 + 4ab + 4b^2) \]
Combine like terms:
\[ = (4a^2 + 2a^2 + a^2) + (4ab + 5ab + 4ab) + (b^2 + 2b^2 + 4b^2) \]
\[ = 7a^2 + 13ab + 7b^2 \]

### Step 4: Write the final factorization
Now we can write the factorization of the original expression:
\[ (2a + b)^3 - (a + 2b)^3 = (a - b)(7a^2 + 13ab + 7b^2) \]

### Final Answer
Thus, the factorization of \( (2a + b)^3 - (a + 2b)^3 \) is:
\[ \boxed{(a - b)(7a^2 + 13ab + 7b^2)} \]