Factorise :
`4xy -x^(2) - 4y^(2) + z^(2)`

Answer

To factorise the expression \(4xy - x^2 - 4y^2 + z^2\), we will follow these steps:

### Step 1: Rearrange the Expression
We can rearrange the expression as follows:
\[ z^2 - x^2 - 4y^2 + 4xy \]

### Step 2: Group Terms
Next, we can group the terms related to \(x\) and \(y\):
\[ z^2 - (x^2 - 4xy + 4y^2) \]

### Step 3: Factor Out the Negative Sign
Now, we can factor out a negative sign from the grouped terms:
\[ z^2 - \left( x^2 - 4xy + 4y^2 \right) = z^2 - \left( (x - 2y)^2 \right) \]

### Step 4: Recognize the Difference of Squares
The expression now resembles the difference of squares:
\[ z^2 - (x - 2y)^2 \]
We can apply the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

### Step 5: Apply the Difference of Squares Formula
Let \(a = z\) and \(b = (x - 2y)\):
\[ (z + (x - 2y))(z - (x - 2y)) \]

### Step 6: Simplify the Factors
Now we simplify the factors:
\[ (z + x - 2y)(z - x + 2y) \]

### Final Answer
Thus, the factorised form of the expression \(4xy - x^2 - 4y^2 + z^2\) is:
\[ (z + x - 2y)(z - x + 2y) \]

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