To solve the sequence \(0, 4, 8, 24, 64, 176, ?\), let's analyze the pattern by checking the relationship between consecutive terms:

1. **First difference** (difference between consecutive terms):
   \[
   4 - 0 = 4, \quad 8 - 4 = 4, \quad 24 - 8 = 16, \quad 64 - 24 = 40, \quad 176 - 64 = 112
   \]
   First difference sequence: \(4, 4, 16, 40, 112\)

2. **Second difference** (difference between consecutive first differences):
   \[
   4 - 4 = 0, \quad 16 - 4 = 12, \quad 40 - 16 = 24, \quad 112 - 40 = 72
   \]
   Second difference sequence: \(0, 12, 24, 72\)

3. **Third difference** (difference between consecutive second differences):
   \[
   12 - 0 = 12, \quad 24 - 12 = 12, \quad 72 - 24 = 48
   \]
   Third difference sequence: \(12, 12, 48\)

The differences suggest that the pattern may involve a higher degree polynomial or a combination of sequences.

Now, let's predict the next term in the first difference sequence using the observed pattern:
- The third difference sequence shows an increasing gap, so we might anticipate a continuation with an additional 48:
  \[
  48 + 72 = 120 \quad \text{(add this to the second difference sequence)}
  \]
- Add this to the last value in the second difference sequence:
  \[
  120 + 112 = 232 \quad \text{(predicted next first difference)}
  \]
- Add this to the last value in the first difference sequence:
  \[
  232 + 176 = 408
  \]

Thus, the next term in the sequence should be:

\[
\boxed{408}
\]

Would you like more details, or do you have any questions? 

Here are some related questions for further exploration:

1. What is the general form of the sequence if modeled as a polynomial?
2. Can this sequence be represented using a recursive formula?
3. How would the sequence change if we used a different starting value?
4. What is the significance of the third differences being equal to 12 initially?
5. Can the sequence be linked to a real-world application or model?

**Tip:** When analyzing sequences, always check differences up to the third or fourth level; this can reveal patterns indicating polynomial behavior.

Explore the solution to the sequence 0, 4, 8, 24, 64, 176, ?, analyzing patterns and predicting the next term. This problem involves sequence analysis, patterns, and polynomial behavior, suitable for advanced high school students.

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