Factorise `: x ^(4) + x ^(2) y ^(2) + y ^(4).`

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ x^4 + x^2y^2 + y^4 \]
We can add and subtract \( x^2y^2 \) to help us factor the expression:
\[ x^4 + 2x^2y^2 + y^4 - x^2y^2 \]

### Step 2: Group the terms
Now we can group the first three terms and the last term:
\[ (x^4 + 2x^2y^2 + y^4) - x^2y^2 \]

### Step 3: Recognize a perfect square
The expression \( x^4 + 2x^2y^2 + y^4 \) can be recognized as a perfect square:
\[ (x^2 + y^2)^2 \]
Thus, we rewrite the expression:
\[ (x^2 + y^2)^2 - x^2y^2 \]

### Step 4: Apply the difference of squares
Now we have a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a + b)(a - b) \):
Let \( a = x^2 + y^2 \) and \( b = xy \). Therefore, we can write:
\[ ((x^2 + y^2) + xy)((x^2 + y^2) - xy) \]

### Step 5: Write the final factorized form
Thus, the factorized form of the expression \( x^4 + x^2y^2 + y^4 \) is:
\[ (x^2 + y^2 + xy)(x^2 + y^2 - xy) \]

### Final Answer:
The factorized expression is:
\[ (x^2 + y^2 + xy)(x^2 + y^2 - xy) \]

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