To simplify the expression \(-2(3 + 8 \div 4)^2\), follow these steps:

1. **Simplify inside the parentheses**:  
   First, handle the division inside the parentheses:
   \[
   8 \div 4 = 2
   \]
   So, the expression becomes:
   \[
   -2(3 + 2)^2
   \]

2. **Add inside the parentheses**:
   \[
   3 + 2 = 5
   \]
   Now the expression is:
   \[
   -2(5)^2
   \]

3. **Square the number**:
   \[
   5^2 = 25
   \]
   So, the expression simplifies to:
   \[
   -2 \times 25
   \]

4. **Multiply**:
   \[
   -2 \times 25 = -50
   \]

Thus, the simplified expression is \(\boxed{-50}\).

Would you like more details or have any questions? 

Here are some related questions you might find helpful:
1. How do you simplify expressions with multiple operations?
2. What is the order of operations (PEMDAS/BODMAS)?
3. How do parentheses affect the calculation in an expression?
4. How do you handle negative numbers in multiplication?
5. What are the rules for exponents in mathematical expressions?

**Tip:** Always follow the order of operations (PEMDAS/BODMAS) to simplify expressions correctly!

Learn how to simplify the expression −2(3 + 8 ÷ 4)^2 with a step-by-step solution. This problem involves applying the order of operations (PEMDAS/BODMAS), exponentiation, and basic arithmetic, suitable for students in Grades 6-8.

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