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!