Let's simplify the expression step by step.

The given expression is:

$16(m^2 - 4)^2 - 4(3n)^2$

Step 1: Expand each term

1. The first part is $16(m^2 - 4)^2$. We need to expand $(m^2 - 4)^2$. $(m^2 - 4)^2 = m^4 - 8m^2 + 16$ So, $16(m^2 - 4)^2 = 16(m^4 - 8m^2 + 16) = 16m^4 - 128m^2 + 256$

2. The second part is $4(3n)^2$. Simplifying: $(3n)^2 = 9n^2$ So, $4(3n)^2 = 4 \times 9n^2 = 36n^2$

Step 2: Combine the results

Now, substitute the expanded terms back into the original expression: $16m^4 - 128m^2 + 256 - 36n^2$

Final Simplified Expression:

$16m^4 - 128m^2 - 36n^2 + 256$

This is the fully simplified form of the expression.

Would you like more details on any specific part of this process?

Related Questions:

1. How do you expand binomials like $(m^2 - 4)^2$ using the distributive property?
2. Can we factor the expression $16m^4 - 128m^2 + 256$?
3. How does the order of operations affect expanding and simplifying expressions?
4. Can we factor the entire expression $16m^4 - 128m^2 - 36n^2 + 256$?
5. What are the applications of binomial expansion in higher mathematics?

Tip: When simplifying complex expressions, expand each term carefully and combine like terms to avoid errors.