Factorise :
`25(2a - b)^(2) - 81 b^(2)`

Answer

To factorise the expression \( 25(2a - b)^2 - 81b^2 \), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ 25(2a - b)^2 - 81b^2 \]

### Step 2: Recognize the difference of squares
Notice that this expression is in the form of \( A^2 - B^2 \), where:
- \( A = 5(2a - b) \)
- \( B = 9b \)

### Step 3: Apply the difference of squares formula
We can use the difference of squares formula, which states that \( A^2 - B^2 = (A - B)(A + B) \). Applying this, we have:
\[ (5(2a - b) - 9b)(5(2a - b) + 9b) \]

### Step 4: Simplify each factor
Now, we simplify each factor:
1. For the first factor:
\[ 5(2a - b) - 9b = 10a - 5b - 9b = 10a - 14b \]
2. For the second factor:
\[ 5(2a - b) + 9b = 10a - 5b + 9b = 10a + 4b \]

### Step 5: Write the factored form
Thus, the expression can be factored as:
\[ (10a - 14b)(10a + 4b) \]

### Step 6: Factor out common terms
We can factor out a common factor from each term:
1. From \( 10a - 14b \), we can factor out 2:
\[ 2(5a - 7b) \]
2. From \( 10a + 4b \), we can factor out 2:
\[ 2(5a + 2b) \]

### Final Factored Form
Thus, the final factored form of the expression is:
\[ 2(5a - 7b) \cdot 2(5a + 2b) = 4(5a - 7b)(5a + 2b) \]