Let's analyze both approaches to ensure clarity.

Expression to factor:

$16(x-y)^2 - 9$

Your teacher's method proceeds as follows:

1. Step 1: Recognize that this is a difference of squares: $16(x-y)^2 - 9 = (4(x-y))^2 - 3^2$

2. Step 2: Apply the difference of squares formula:
   $a^2 - b^2 = (a+b)(a-b)$ So, [ (4(x-y))^2 - 3^2 = [4(x-y) + 3][4(x-y) - 3] ]

Your teacher's solution is correct, and the factorization is: [ [4(x-y) + 3][4(x-y) - 3] ]

Now, let's check your method:

You wrote:

1. Step 1:
   $16(x-y)^2 - 9 = 4^2(x-y)^2 - 3^2$ This step is fine, as you are recognizing that both terms are squares.

2. Step 2:
   $4^2(x+y)(x-y) - 3^2$ This step is incorrect because you are factoring $(x-y)^2$ as $(x+y)(x-y)$, which is not valid. The expression $(x-y)^2$ should remain as $(x-y)(x-y)$, or $(x-y)^2$, without turning into a sum or difference of terms.

Thus, your approach contains an error.

Correct Conclusion:

Your teacher's solution is the correct factorization: [ 16(x-y)^2 - 9 = [4(x-y) + 3][4(x-y) - 3] ] Your method deviated because you incorrectly factored $(x-y)^2$.

Would you like a further breakdown or practice on factoring? Here are some related questions:

1. Can you factor $25x^2 - 36$?
2. What is the factorization of $9a^2 - 1$?
3. How would you factor $49y^2 - 64$?
4. Factor $81z^2 - 4$.
5. What is the factored form of $4m^2 - 25$?

Tip: Always be cautious when factoring expressions involving powers, and remember that only perfect squares can be factored using the difference of squares formula.