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

Answer

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

### Step 1: Rearrange the Expression
We start by rearranging the expression to group the terms in a more manageable way:
\[ a^2 - 4ac + 4c^2 - b^2 \]

### Step 2: Recognize a Perfect Square
Next, we observe the first three terms \( a^2 - 4ac + 4c^2 \). This can be recognized as a perfect square:
\[ a^2 - 4ac + 4c^2 = (a - 2c)^2 \]
So we can rewrite the expression as:
\[ (a - 2c)^2 - b^2 \]

### Step 3: Apply the Difference of Squares Formula
Now we have the expression in the form of a difference of squares, which is \( x^2 - y^2 \). We can use the difference of squares formula:
\[ x^2 - y^2 = (x + y)(x - y) \]
In our case, let \( x = (a - 2c) \) and \( y = b \). Thus, we can write:
\[ (a - 2c)^2 - b^2 = (a - 2c + b)(a - 2c - b) \]

### Step 4: Write the Final Factorized Form
Substituting back the values of \( x \) and \( y \), we get the final factorized form:
\[ (a - 2c + b)(a - 2c - b) \]

### Final Answer
The factorized form of the expression \( a^2 - b^2 - 4ac + 4c^2 \) is:
\[ (a - 2c + b)(a - 2c - b) \]

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