Question 1209694
Here's how to determine f(5) using the method of finite differences:

1. **Set up a difference table:**

| x | f(x) | Δf(x) | Δ²f(x) | Δ³f(x) | Δ⁴f(x) |
|---|---|---|---|---|---|
| 0 | 1 |  |  |  |  |
| 1 | 2 | 1 |  |  |  |
| 2 | -7 | -9 | -10 |  |  |
| 3 | 0 | 7 | 16 | 26 |  |
| 4 | 3 | 3 | -4 | -20 | -46 |

Where Δf(x) represents the first difference, Δ²f(x) the second difference, and so on.  Each difference is calculated by subtracting the previous value from the current value. For example, Δf(1) = f(1) - f(0) = 2 - 1 = 1, and Δ²f(2) = Δf(2) - Δf(1) = -9 - 1 = -10.

2. **Since f(x) is a polynomial of degree 4, the fourth differences are constant.**  We can use this fact to find the next values in the table.  The last entry in the Δ⁴f(x) column is -46.

3. **Extend the table:**

We can extend the table by working backwards.
* Δ³f(4) = Δ³f(3) + Δ⁴f(3) = -20 + (-46) = -66
* Δ²f(5) = Δ²f(4) + Δ³f(4) = -4 + (-66) = -70
* Δf(5) = Δf(4) + Δ²f(5) = 3 + (-70) = -67
* f(5) = f(4) + Δf(5) = 3 + (-67) = -64

Therefore, f(5) = -64.

Final Answer: The final answer is $\boxed{-64}$