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.

Learn how to simplify the algebraic expression 16(m^2 - 4)^2 - 4(3n)^2 through binomial expansion and combination of like terms. This detailed step-by-step solution covers key algebraic techniques like expanding binomials and simplifying complex expressions, suitable for Grades 9-11.

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