Factorise the following :
`3a^(2)+7ab -6b^(2)`

Answer

To factorise the expression \(3a^2 + 7ab - 6b^2\), we will use the method of middle term splitting. Here are the steps:

### Step 1: Identify the coefficients
The expression is \(3a^2 + 7ab - 6b^2\). The coefficients are:
- \(A = 3\) (coefficient of \(a^2\))
- \(B = 7\) (coefficient of \(ab\))
- \(C = -6\) (coefficient of \(b^2\))

### Step 2: Find two numbers that multiply to \(A \cdot C\) and add to \(B\)
We need to find two numbers that multiply to \(3 \cdot (-6) = -18\) and add up to \(7\). The numbers that satisfy this condition are \(9\) and \(-2\).

### Step 3: Rewrite the middle term using the two numbers
We can rewrite \(7ab\) as \(9ab - 2ab\). So, the expression becomes:
\[ 3a^2 + 9ab - 2ab - 6b^2 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (3a^2 + 9ab) + (-2ab - 6b^2) \]

### Step 5: Factor out the common factors from each group
From the first group \(3a^2 + 9ab\), we can factor out \(3a\):
\[ 3a(a + 3b) \]
From the second group \(-2ab - 6b^2\), we can factor out \(-2b\):
\[ -2b(a + 3b) \]

### Step 6: Combine the factored terms
Now we have:
\[ 3a(a + 3b) - 2b(a + 3b) \]
We can see that \((a + 3b)\) is a common factor:
\[ (a + 3b)(3a - 2b) \]

### Final Answer
Thus, the factorised form of \(3a^2 + 7ab - 6b^2\) is:
\[ (a + 3b)(3a - 2b) \]

---