Let's solve the expression step by step:

Given expression: $-4^2 + 8(4) + 16$

1. Evaluate $-4^2$:

   It's important to note that $-4^2$ means $-(4^2)$, not $(-4)^2$. So: $-4^2 = -(4 \times 4) = -16$

   Now the expression is: $-16 + 8(4) + 16$

2. Evaluate $8(4)$: $8 \times 4 = 32$

   Now the expression becomes: $-16 + 32 + 16$

3. Add the numbers together: $-16 + 32 = 16$ $16 + 16 = 32$

So, the final result is: $32$

Do you want more details on any of these steps or have any questions?

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Here are five related questions you might find interesting:

1. How does the interpretation of the negative sign before an exponent affect the result?
2. What is the difference between $-4^2$ and $(-4)^2$?
3. How would the expression change if we used $(-4)^2$ instead of $-4^2$?
4. What happens to the expression if you change the multiplication to addition inside the parentheses?
5. How can this approach be applied to more complex expressions with multiple operations?

Tip: Always remember that exponents are applied before multiplication or negation when following the order of operations.