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!