https://file.aiquickdraw.com/mathbot/file/1725093604027iuw1igqm.jpg

The expression in the image is \( 49a^2b^2 - 64b^2 \).

To factor this expression, follow these steps:

1. **Factor out the common factor:** 
   - Both terms have a common factor of \( b^2 \). So, we can factor out \( b^2 \) from the expression.
   \[
   49a^2b^2 - 64b^2 = b^2(49a^2 - 64)
   \]

2. **Factor the difference of squares:**
   - The expression inside the parentheses, \( 49a^2 - 64 \), is a difference of squares.
   - The difference of squares formula is \( x^2 - y^2 = (x - y)(x + y) \).
   - Here, \( 49a^2 \) is \( (7a)^2 \), and \( 64 \) is \( 8^2 \).

   Therefore:
   \[
   49a^2 - 64 = (7a - 8)(7a + 8)
   \]

3. **Combine the factors:**
   \[
   49a^2b^2 - 64b^2 = b^2(7a - 8)(7a + 8)
   \]

So, the fully factored form of the expression is \( b^2(7a - 8)(7a + 8) \).

Would you like further details on any part of this process, or do you have any other questions?

Here are some related questions:
1. What is the difference of squares, and how do you recognize it?
2. Why is factoring out the greatest common factor (GCF) the first step in factoring?
3. How would you factor a trinomial that is not a perfect square?
4. Can you apply the difference of squares method to factor higher-degree polynomials?
5. What are some practical applications of factoring in mathematics?

**Tip:** Recognizing patterns like the difference of squares can make factoring much quicker and easier. Practice identifying these patterns in different expressions.

Learn how to factor the expression 49a^2b^2 - 64b^2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring and the difference of squares formula. Suitable for students in Grades 9-12.

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