Factorise :
`4a^(2) - 12a + 9 - 49b^(2)`

Answer

To factorise the expression \( 4a^2 - 12a + 9 - 49b^2 \), we will follow these steps:

### Step 1: Rearranging the Expression
We start with the expression:
\[ 4a^2 - 12a + 9 - 49b^2 \]
We can group the first three terms together and the last term separately:
\[ (4a^2 - 12a + 9) - 49b^2 \]

### Step 2: Recognising a Perfect Square
Next, we will focus on the quadratic expression \( 4a^2 - 12a + 9 \). We can rewrite this as:
\[ (2a)^2 - 2 \cdot (2a) \cdot 3 + 3^2 \]
This is a perfect square trinomial, which can be factored as:
\[ (2a - 3)^2 \]

### Step 3: Rewriting the Expression
Now we can rewrite the original expression using our factorization:
\[ (2a - 3)^2 - 49b^2 \]

### Step 4: Applying the Difference of Squares Formula
Now we have a difference of squares, which can be factored using the identity \( x^2 - y^2 = (x + y)(x - y) \). Here, let \( x = (2a - 3) \) and \( y = 7b \):
\[ (2a - 3)^2 - (7b)^2 = \left((2a - 3) + 7b\right)\left((2a - 3) - 7b\right) \]

### Step 5: Final Factorised Form
Thus, the factorised form of the expression \( 4a^2 - 12a + 9 - 49b^2 \) is:
\[ (2a - 3 + 7b)(2a - 3 - 7b) \]

### Summary of the Factorisation
The final answer is:
\[ (2a - 3 + 7b)(2a - 3 - 7b) \]