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The expression given is:

\[
5(-m + 2n)(-m - 2n)
\]

Let's break this down step by step:

### Step 1: Expand the two binomials
First, we need to expand the product of \((-m + 2n)(-m - 2n)\).

We'll use the distributive property (FOIL method):
\[
(-m + 2n)(-m - 2n) = (-m)(-m) + (-m)(-2n) + (2n)(-m) + (2n)(-2n)
\]

Now, compute each term:

\[
(-m)(-m) = m^2
\]
\[
(-m)(-2n) = 2mn
\]
\[
(2n)(-m) = -2mn
\]
\[
(2n)(-2n) = -4n^2
\]

Now combine all terms:
\[
m^2 + 2mn - 2mn - 4n^2
\]

Notice that the \( 2mn \) and \(-2mn\) cancel each other out:
\[
m^2 - 4n^2
\]

### Step 2: Multiply by 5
Now, multiply the result by 5:
\[
5(m^2 - 4n^2) = 5m^2 - 20n^2
\]

### Final Answer:
\[
5m^2 - 20n^2
\]

Would you like further details or clarifications on any part of the solution? Here are five follow-up questions to consider:

1. How would you approach expanding different binomials, like \( (m + n)(m - n) \)?
2. What happens if we distribute a negative number through a binomial product?
3. Can you derive the result of \( (x + y)^2 \) using the same technique?
4. What is the geometric interpretation of expanding products like \( (a + b)(a - b) \)?
5. How do special factorizations (like difference of squares) simplify the expansion process?

**Tip:** The difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \) is a useful tool to recognize patterns in binomial products!

Learn how to expand and simplify the expression 5(-m + 2n)(-m - 2n) using binomial expansion and the difference of squares formula. This problem helps reinforce key algebraic skills, suitable for students in Grades 9-11.

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