Factorise: `4a^2 + 12ab + 9b^2 - 8a - 12b `

Options

A. (2a - 3b) (2a - 3b - 4)
B. (2a + 3b) (2a + 3b - 4)
C. (2a - 3b) (2a + 3b - 2)
D. (2a + 3b) (3a + 2b - 4)

Answer

To factorise the expression \(4a^2 + 12ab + 9b^2 - 8a - 12b\), we can follow these steps:

### Step 1: Rearrange the expression
We start with the expression:
\[ 4a^2 + 12ab + 9b^2 - 8a - 12b \]

### Step 2: Group the terms
We can group the terms in a way that makes it easier to factor:
\[ (4a^2 + 12ab + 9b^2) + (-8a - 12b) \]

### Step 3: Factor the first group
The first group \(4a^2 + 12ab + 9b^2\) can be factored as a perfect square:
\[ (2a + 3b)^2 \]
because \(4a^2\) is \((2a)^2\), \(9b^2\) is \((3b)^2\), and \(12ab\) is \(2 \cdot 2a \cdot 3b\).

### Step 4: Factor the second group
Now, we can factor out common terms from the second group:
\[ -8a - 12b = -4(2a + 3b) \]

### Step 5: Combine the factored terms
Now we can rewrite the expression using the factored forms:
\[ (2a + 3b)^2 - 4(2a + 3b) \]

### Step 6: Let \(x = 2a + 3b\)
To simplify the expression, we can let \(x = 2a + 3b\). The expression now becomes:
\[ x^2 - 4x \]

### Step 7: Factor the quadratic expression
We can factor this quadratic expression:
\[ x(x - 4) \]

### Step 8: Substitute back for \(x\)
Now, substituting back \(x = 2a + 3b\), we have:
\[ (2a + 3b)(2a + 3b - 4) \]

### Final Answer
Thus, the factorised form of the expression \(4a^2 + 12ab + 9b^2 - 8a - 12b\) is:
\[ (2a + 3b)(2a + 3b - 4) \]