Factorise :
`9a^(2) + 3a - 8b - 64b^(2)`

Answer

To factorise the expression \(9a^2 + 3a - 8b - 64b^2\), we can follow these steps:

### Step 1: Rearrange the expression
Rearranging the terms can sometimes help in identifying common factors. We can write the expression as:
\[ 9a^2 + 3a - (8b + 64b^2) \]

### Step 2: Factor out common terms
Notice that \(8b + 64b^2\) can be factored. We can factor out \(8b\):
\[ 9a^2 + 3a - 8b(1 + 8b) \]

### Step 3: Rewrite \(9a^2 + 3a\)
Next, we can factor out \(3a\) from the first two terms:
\[ 3a(3a + 1) - 8b(1 + 8b) \]

### Step 4: Recognize the difference of squares
Now, we can rewrite the expression in a form that resembles the difference of squares. We can express \(9a^2\) as \((3a)^2\) and \(64b^2\) as \((8b)^2\):
\[ (3a)^2 + 3a - (8b)^2 \]

### Step 5: Use the identity \(a^2 - b^2 = (a + b)(a - b)\)
We can apply the difference of squares identity:
\[ (3a + 8b)(3a - 8b) + 3a \]

### Step 6: Combine like terms
Now we can combine the terms:
\[ (3a - 8b)(3a + 8b) + 3a \]

### Step 7: Final factorization
Finally, we can factor out \(3a - 8b\):
\[ (3a - 8b)(3a + 8b + 1) \]

Thus, the factorized form of the expression \(9a^2 + 3a - 8b - 64b^2\) is:
\[ (3a - 8b)(3a + 8b + 1) \]