factorize : `25x^2-10x+1-36y^2`

Answer

To factorize the expression \( 25x^2 - 10x + 1 - 36y^2 \), we can follow these steps:

### Step 1: Rearrange the expression
We start with the expression:
\[ 25x^2 - 10x + 1 - 36y^2 \]
We can group the first three terms together and keep the last term separate:
\[ (25x^2 - 10x + 1) - 36y^2 \]

### Step 2: Recognize a perfect square
The first part \( 25x^2 - 10x + 1 \) can be recognized as a perfect square. We can rewrite it as:
\[ (5x)^2 - 2(5x)(1) + 1^2 \]
This matches the format \( a^2 - 2ab + b^2 \), which is \( (a - b)^2 \). Therefore, we can express it as:
\[ (5x - 1)^2 \]

### Step 3: Rewrite the expression
Now, substituting back into our expression, we have:
\[ (5x - 1)^2 - 36y^2 \]

### Step 4: Recognize the difference of squares
The expression \( (5x - 1)^2 - 36y^2 \) is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a + b)(a - b) \). Here, we can let:
- \( a = (5x - 1) \)
- \( b = 6y \)

Thus, we can write:
\[ (5x - 1 + 6y)(5x - 1 - 6y) \]

### Step 5: Final factorized form
Putting it all together, the factorized form of the expression \( 25x^2 - 10x + 1 - 36y^2 \) is:
\[ (5x + 6y - 1)(5x - 6y - 1) \]

### Summary of the factorization:
The final answer is:
\[ (5x + 6y - 1)(5x - 6y - 1) \]