Factorise : (i) `1-4 (a-2b)^2`

Answer

To factorise the expression \(1 - 4(a - 2b)^2\), we will follow these steps:

### Step 1: Rewrite the Expression
Start with the given expression:
\[ 1 - 4(a - 2b)^2 \]

### Step 2: Recognize the Structure
Notice that \(4(a - 2b)^2\) can be rewritten as \((2(a - 2b))^2\). Thus, we can express the equation as:
\[ 1 - (2(a - 2b))^2 \]

### Step 3: Apply the Difference of Squares Formula
We can use the difference of squares formula, which states that \(x^2 - y^2 = (x + y)(x - y)\). Here, let \(x = 1\) and \(y = 2(a - 2b)\):
\[ 1 - (2(a - 2b))^2 = (1 + 2(a - 2b))(1 - 2(a - 2b)) \]

### Step 4: Simplify Each Factor
Now simplify each factor:
1. For the first factor:
\[ 1 + 2(a - 2b) = 1 + 2a - 4b \]
2. For the second factor:
\[ 1 - 2(a - 2b) = 1 - 2a + 4b \]

### Step 5: Write the Final Factorised Form
Putting it all together, the factorised form of the expression is:
\[ (1 + 2a - 4b)(1 - 2a + 4b) \]

### Final Answer
Thus, the factorised form of \(1 - 4(a - 2b)^2\) is:
\[ (1 + 2a - 4b)(1 - 2a + 4b) \]

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