Factorise:
` 4(2a - b+c) ^(2) - 9 ( a+b- c) ^(2)`

Answer

To factorise the expression \( 4(2a - b + c)^2 - 9(a + b - c)^2 \), we can follow these steps:

### Step 1: Identify the structure
Notice that the expression is in the form of \( A^2 - B^2 \), where:
- \( A = \sqrt{4}(2a - b + c) = 2(2a - b + c) \)
- \( B = \sqrt{9}(a + b - c) = 3(a + b - c) \)

### Step 2: Apply the difference of squares formula
Recall the difference of squares formula:
\[ A^2 - B^2 = (A + B)(A - B) \]
Using this, we can rewrite our expression as:
\[ (2(2a - b + c) + 3(a + b - c))(2(2a - b + c) - 3(a + b - c)) \]

### Step 3: Simplify \( A + B \)
Now, let's simplify \( A + B \):
\[ A + B = 2(2a - b + c) + 3(a + b - c) \]
Distributing the terms:
\[ = 4a - 2b + 2c + 3a + 3b - 3c \]
Combining like terms:
\[ = (4a + 3a) + (-2b + 3b) + (2c - 3c) = 7a + b - c \]

### Step 4: Simplify \( A - B \)
Next, simplify \( A - B \):
\[ A - B = 2(2a - b + c) - 3(a + b - c) \]
Distributing the terms:
\[ = 4a - 2b + 2c - 3a - 3b + 3c \]
Combining like terms:
\[ = (4a - 3a) + (-2b - 3b) + (2c + 3c) = a - 5b + 5c \]

### Step 5: Write the final factorized form
Now, we can write the complete factorized form:
\[ (7a + b - c)(a - 5b + 5c) \]

### Final Answer
Thus, the factorized form of the expression \( 4(2a - b + c)^2 - 9(a + b - c)^2 \) is:
\[ (7a + b - c)(a - 5b + 5c) \]