We can simplify the given expression step by step:

### Expression:
\[
(x - 2y)(x + 2y) - (2x - y)(2x + y)
\]

### Step 1: Apply the difference of squares formula
The first part \((x - 2y)(x + 2y)\) is a difference of squares, which simplifies as:
\[
(x - 2y)(x + 2y) = x^2 - (2y)^2 = x^2 - 4y^2
\]

Similarly, apply the difference of squares formula to the second part \((2x - y)(2x + y)\):
\[
(2x - y)(2x + y) = (2x)^2 - y^2 = 4x^2 - y^2
\]

### Step 2: Substitute the simplified forms back into the expression
Now we have:
\[
(x^2 - 4y^2) - (4x^2 - y^2)
\]

### Step 3: Distribute the subtraction
\[
x^2 - 4y^2 - 4x^2 + y^2
\]

### Step 4: Combine like terms
\[
(x^2 - 4x^2) + (-4y^2 + y^2) = -3x^2 - 3y^2
\]

### Final Answer:
\[
-3x^2 - 3y^2
\]

Would you like more details or have any other questions?

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Here are 5 related questions to expand on this:

1. Can you find the value of the expression when \( x = 1 \) and \( y = 2 \)?
2. How would the expression change if the signs inside the second term were reversed?
3. Can you factor the expression \(-3x^2 - 3y^2\) further?
4. What happens if you substitute \( x = y \) in the original expression?
5. How would the solution differ if one of the terms was a sum of cubes instead of squares?

**Tip**: When simplifying complex algebraic expressions, applying known formulas like the difference of squares can make the process quicker and easier.

Learn how to simplify the expression (x - 2y)(x + 2y) - (2x - y)(2x + y) by applying the difference of squares formula. This step-by-step solution covers key algebraic concepts, perfect for high school students in Grades 9-12.

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