Factorise :
` 4a^(2) - 4ab+ b^(2) - 4x^(2)`

Answer

To factorize the expression \( 4a^2 - 4ab + b^2 - 4x^2 \), we can follow these steps:

### Step 1: Group the terms
We start with the expression:
\[ 4a^2 - 4ab + b^2 - 4x^2 \]
We can rearrange it to:
\[ (4a^2 - 4ab + b^2) - 4x^2 \]

### Step 2: Recognize a perfect square
The first part \( 4a^2 - 4ab + b^2 \) can be recognized as a perfect square trinomial. We can rewrite it as:
\[ (2a - b)^2 \]
Thus, we have:
\[ (2a - b)^2 - 4x^2 \]

### Step 3: Apply the difference of squares formula
Now, we have a difference of squares:
\[ (2a - b)^2 - (2x)^2 \]
We can apply the difference of squares formula, which states that \( A^2 - B^2 = (A + B)(A - B) \). Here, \( A = (2a - b) \) and \( B = 2x \).

### Step 4: Factor using the difference of squares
Using the formula, we can factor the expression as follows:
\[ (2a - b + 2x)(2a - b - 2x) \]

### Final Result
Thus, the factorized form of the expression \( 4a^2 - 4ab + b^2 - 4x^2 \) is:
\[ (2a - b + 2x)(2a - b - 2x) \]

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