factorize the given expression `9a^2+6a+1-36b^2`

Answer

To factor the expression \( 9a^2 + 6a + 1 - 36b^2 \), we will follow these steps:

### Step 1: Group the first three terms
We start with the expression:
\[ 9a^2 + 6a + 1 - 36b^2 \]
We can focus on the first three terms \( 9a^2 + 6a + 1 \).

### Step 2: Recognize the perfect square
Notice that \( 9a^2 + 6a + 1 \) can be rewritten as:
\[ (3a)^2 + 2 \cdot (3a) \cdot 1 + 1^2 \]
This is in the form of \( a^2 + 2ab + b^2 \), which factors to \( (a + b)^2 \). Here, \( a = 3a \) and \( b = 1 \).

### Step 3: Factor the perfect square
Thus, we can write:
\[ 9a^2 + 6a + 1 = (3a + 1)^2 \]

### Step 4: Rewrite the expression
Now, substitute this back into the original expression:
\[ (3a + 1)^2 - 36b^2 \]

### Step 5: Recognize the difference of squares
The expression \( (3a + 1)^2 - 36b^2 \) is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = (3a + 1) \) and \( b = 6b \).

### Step 6: Apply the difference of squares formula
Applying the difference of squares formula, we get:
\[ (3a + 1 - 6b)(3a + 1 + 6b) \]

### Final Answer
Thus, the factorized form of the expression \( 9a^2 + 6a + 1 - 36b^2 \) is:
\[ (3a + 1 - 6b)(3a + 1 + 6b) \]

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