Factorise :
`4x^(4) + 9y^(4) + 11x^(2)y^(2)`

Answer

To factorise the expression \(4x^4 + 9y^4 + 11x^2y^2\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 4x^4 + 9y^4 + 11x^2y^2 \]

### Step 2: Recognize squares
Notice that \(4x^4\) can be rewritten as \((2x^2)^2\) and \(9y^4\) can be rewritten as \((3y^2)^2\). Thus, we can rewrite the expression as:
\[ (2x^2)^2 + (3y^2)^2 + 11x^2y^2 \]

### Step 3: Rearranging the terms
Next, we can rearrange the terms in a way that helps us factor:
\[ (2x^2)^2 + (3y^2)^2 + 2(2x^2)(3y^2) - x^2y^2 \]
Here, we have added and subtracted \(x^2y^2\) to maintain equality.

### Step 4: Grouping the terms
Now, we can group the first three terms:
\[ (2x^2 + 3y^2)^2 - (xy)^2 \]

### Step 5: Apply the difference of squares
Now we can apply the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\). Here, let \(a = (2x^2 + 3y^2)\) and \(b = xy\):
\[ (2x^2 + 3y^2 + xy)(2x^2 + 3y^2 - xy) \]

### Final Result
Thus, the factorised form of the expression \(4x^4 + 9y^4 + 11x^2y^2\) is:
\[ (2x^2 + 3y^2 + xy)(2x^2 + 3y^2 - xy) \]

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