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

Answer

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

### Step 1: Recognize the Structure
The expression resembles the difference of squares. We can rewrite it in the form \(a^2 - b^2\) where:
- \(a = (x^2 + y^2 - z^2)\)
- \(b = 2xy\)

### Step 2: Apply the Difference of Squares Formula
Using the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\), we can factor the expression:
\[ (x^2 + y^2 - z^2)^2 - (2xy)^2 = (x^2 + y^2 - z^2 + 2xy)(x^2 + y^2 - z^2 - 2xy) \]

### Step 3: Simplify Each Factor
Now, we need to simplify each factor:
1. **First Factor**:
\[ x^2 + y^2 - z^2 + 2xy = (x^2 + 2xy + y^2) - z^2 = (x + y)^2 - z^2 \]
This can be factored further using the difference of squares:
\[ (x + y + z)(x + y - z) \]

2. **Second Factor**:
\[ x^2 + y^2 - z^2 - 2xy = (x^2 - 2xy + y^2) - z^2 = (x - y)^2 - z^2 \]
This can also be factored further using the difference of squares:
\[ (x - y + z)(x - y - z) \]

### Step 4: Combine the Factors
Putting it all together, we have:
\[ (x + y + z)(x + y - z)(x - y + z)(x - y - z) \]

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