Fractorise:
`2a + 6b - 3 (a + 3b)^(2)`

Answer

To factorize the expression \( 2a + 6b - 3(a + 3b)^2 \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the original expression:
\[ 2a + 6b - 3(a + 3b)^2 \]

### Step 2: Expand the square
Expand \( (a + 3b)^2 \):
\[ (a + 3b)^2 = a^2 + 6ab + 9b^2 \]
Now substitute this back into the expression:
\[ 2a + 6b - 3(a^2 + 6ab + 9b^2) \]

### Step 3: Distribute the -3
Distribute \(-3\) across the expanded square:
\[ 2a + 6b - 3a^2 - 18ab - 27b^2 \]

### Step 4: Rearrange the expression
Rearranging the terms gives:
\[ -3a^2 + 2a - 18ab + 6b - 27b^2 \]

### Step 5: Factor out common terms
Now, we can group the terms:
\[ -3(a^2 + 6ab + 9b^2) + 2a + 6b \]
Notice that \(a + 3b\) is common in the quadratic part. We can factor it out:
\[ -3(a + 3b)(a + 3b) + 2a + 6b \]

### Step 6: Combine like terms
Now, we can factor \(a + 3b\) out of the entire expression:
\[ = (a + 3b)(2 - 3(a + 3b)) \]

### Step 7: Final expression
Thus, the factorized form of the expression is:
\[ (a + 3b)(2 - 3(a + 3b)) \]