Factorise by grouping method :
`a^(2) + 4b^(2) - 3a + 6b - 4ab`

Answer

To factorise the expression \( a^2 + 4b^2 - 3a + 6b - 4ab \) by the grouping method, we can follow these steps:

### Step 1: Rearrange the expression
We start by rearranging the terms in the expression for better grouping:
\[ a^2 - 3a - 4ab + 4b^2 + 6b \]

### Step 2: Group the terms
Next, we group the terms into two parts:
\[ (a^2 - 4ab - 3a) + (4b^2 + 6b) \]

### Step 3: Factor out common terms
Now, we can factor out common terms from each group:
1. From the first group \( a^2 - 4ab - 3a \), we can factor out \( a \):
\[ a(a - 4b - 3) \]
2. From the second group \( 4b^2 + 6b \), we can factor out \( 2b \):
\[ 2b(2b + 3) \]

### Step 4: Rewrite the expression
Now we rewrite the expression as:
\[ a(a - 4b - 3) + 2b(2b + 3) \]

### Step 5: Recognize the perfect square
We can recognize that \( a^2 - 4ab + 4b^2 \) can be rewritten as a perfect square:
\[ (a - 2b)^2 \]
Thus, we can rewrite the entire expression:
\[ (a - 2b)^2 - 3(a - 2b) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (a - 2b) \):
\[ (a - 2b)((a - 2b) - 3) \]
This simplifies to:
\[ (a - 2b)(a - 2b - 3) \]

### Final Answer
The factorised form of the expression \( a^2 + 4b^2 - 3a + 6b - 4ab \) is:
\[ (a - 2b)(a - 2b - 3) \]