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The question asks to find which polynomial sequence shows constant second differences.

To check this, we calculate the first and second differences of each sequence:

1. **Sequence {0, 4, 16, 34}:**
   - First differences: \(4 - 0 = 4\), \(16 - 4 = 12\), \(34 - 16 = 18\)
   - Second differences: \(12 - 4 = 8\), \(18 - 12 = 6\) (not constant)

2. **Sequence {1, 7, 12, 19}:**
   - First differences: \(7 - 1 = 6\), \(12 - 7 = 5\), \(19 - 12 = 7\)
   - Second differences: \(5 - 6 = -1\), \(7 - 5 = 2\) (not constant)

3. **Sequence {2, 6, 11, 14}:**
   - First differences: \(6 - 2 = 4\), \(11 - 6 = 5\), \(14 - 11 = 3\)
   - Second differences: \(5 - 4 = 1\), \(3 - 5 = -2\) (not constant)

4. **Sequence {-6, -1, 14, 39}:**
   - First differences: \(-1 - (-6) = 5\), \(14 - (-1) = 15\), \(39 - 14 = 25\)
   - Second differences: \(15 - 5 = 10\), \(25 - 15 = 10\) (constant)

Thus, the sequence with constant second differences is **{-6, -1, 14, 39}**.

Do you want further details or have any questions?

Here are 5 related questions to expand on:
1. What are first and second differences in polynomial sequences?
2. How do constant second differences indicate a quadratic polynomial?
3. Can a sequence with constant first differences also have constant second differences?
4. How can higher-order differences (third, fourth, etc.) be used to identify polynomials?
5. Why does a sequence with constant second differences represent a quadratic function?

**Tip:** Constant second differences indicate a quadratic relationship in sequences, just like constant first differences indicate a linear one.

Which polynomial sequence shows that the terms of the 2nd differences are constant?

This problem involves identifying a polynomial sequence from a list that shows constant second differences. The solution covers first and second differences and explains how constant second differences indicate a quadratic sequence. Suitable for students in Grades 8-10.

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