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.