The factors of `(a^(2)+36b^(2))^(2)-169a^(2)b^(2)` are:

Options

A. `(a+13b)(a-13b)`
B. `(a+4b)(a+9b)(a-4b)(a-9b)`
C. `(a+6b)(a-13b)(a+13b)(a+6b)`
D. `(a-13b)(a+6b)`

Answer

To find the factors of the expression \((a^2 + 36b^2)^2 - 169a^2b^2\), we can follow these steps:

### Step 1: Recognize the Structure
We can see that the expression resembles a difference of squares. We can rewrite it as:
\[ (A^2 - B^2) \]
where \(A = (a^2 + 36b^2)\) and \(B = 13ab\). Thus, we have:
\[ (A^2 - B^2) = (a^2 + 36b^2)^2 - (13ab)^2 \]

### Step 2: Apply the Difference of Squares Formula
Using the difference of squares identity, \(X^2 - Y^2 = (X + Y)(X - Y)\), we can factor the expression:
\[ (a^2 + 36b^2 + 13ab)(a^2 + 36b^2 - 13ab) \]

### Step 3: Simplify Each Factor
Now we need to simplify each of the factors:

1. **First Factor:**
\[ a^2 + 36b^2 + 13ab \]

2. **Second Factor:**
\[ a^2 + 36b^2 - 13ab \]

### Step 4: Factor Further if Possible
Next, we can check if these factors can be factored further.

For the first factor \(a^2 + 36b^2 + 13ab\):
- This can be viewed as a quadratic in terms of \(a\):
\[ a^2 + 13ab + 36b^2 \]
To factor this, we look for two numbers that multiply to \(36b^2\) and add to \(13b\). These numbers are \(9b\) and \(4b\):
\[ (a + 9b)(a + 4b) \]

For the second factor \(a^2 + 36b^2 - 13ab\):
- This can also be viewed as a quadratic in terms of \(a\):
\[ a^2 - 13ab + 36b^2 \]
To factor this, we look for two numbers that multiply to \(36b^2\) and add to \(-13b\). These numbers are \(-9b\) and \(-4b\):
\[ (a - 9b)(a - 4b) \]

### Final Factored Form
Putting it all together, we have:
\[ (a + 9b)(a + 4b)(a - 9b)(a - 4b) \]

### Conclusion
Thus, the complete factorization of the expression \((a^2 + 36b^2)^2 - 169a^2b^2\) is:
\[ (a + 9b)(a + 4b)(a - 9b)(a - 4b) \]