An expression is F(x) = ax4 - bx2 + x + 5 and F(-3) = 2. Find F(3).

For an expression built from even powers of x plus a linear term plus a constant — such as ax4 - bx2 + x + 5 — replacing x by -x leaves every even-power term unchanged and flips the sign of only the odd-power (linear) term. So F(-3) and F(3) share the same even part 81a - 9b, differing only in how the linear term contributes.

1. Substitute x = -3: F(-3) = a(-3)4 - b(-3)2 + (-3) + 5 = 81a - 9b + 2.

2. Use the given F(-3) = 2: 81a - 9b + 2 = 2, so 81a - 9b = 0.

3. Substitute x = 3: F(3) = a(3)4 - b(3)2 + 3 + 5 = 81a - 9b + 8.

4. Since 81a - 9b = 0: F(3) = 0 + 8 = 8.

Independent check: adding the two evaluations gives F(3) + F(-3) = 2(81a - 9b) + 10 = 2(0) + 10 = 10. Since F(-3) = 2, this gives F(3) = 10 - 2 = 8, confirming the result.

Therefore, F(3) = 8.