Factorise :
`4x^(2) - 12ax - y^(2) - z^(2) - 2yz + 9a^(2)`

Answer

To factorise the expression \( 4x^2 - 12ax - y^2 - z^2 - 2yz + 9a^2 \), we can follow these steps:

### Step 1: Rearranging the Terms
We start by rearranging the terms in the expression:
\[ 4x^2 - 12ax + 9a^2 - (y^2 + z^2 + 2yz) \]

### Step 2: Recognizing Perfect Squares
Next, we can identify the perfect squares in the expression:
- \( 4x^2 \) can be written as \( (2x)^2 \)
- \( 9a^2 \) can be written as \( (3a)^2 \)
- The expression \( y^2 + z^2 + 2yz \) can be recognized as \( (y + z)^2 \)

So we rewrite the expression:
\[ (2x)^2 - 12ax + (3a)^2 - (y + z)^2 \]

### Step 3: Completing the Square
Now, we can complete the square for the first part:
\[ (2x - 3a)^2 - (y + z)^2 \]

### Step 4: Applying the Difference of Squares Formula
Now we have a difference of squares, which can be factored using the identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ (2x - 3a - (y + z))(2x - 3a + (y + z)) \]

### Step 5: Final Factorization
Thus, the final factorization of the expression is:
\[ (2x - 3a - y - z)(2x - 3a + y + z) \]

### Summary of the Factorization
The factorized form of \( 4x^2 - 12ax - y^2 - z^2 - 2yz + 9a^2 \) is:
\[ (2x - 3a - y - z)(2x - 3a + y + z) \]