f(x+1) = 2(x+1)^3 - 5(x+1) + 6
= 2(x^3 + 3x^2 + 3x + 1) - 5x - 5 + 6
= 2x^3 + 6x^2 + 6x + 2 - 5x + 1
= 2x^3 + 6x^2 + x + 3
f(x) = 2x^3 - 5x + 6
f(x+1) - f(x) = 2x^3 + 6x^2 + x + 3 - (2x^3 - 5x + 6)
= 2x^3 + 6x^2 + x + 3 - 2x^3 + 5x - 6
= 6x^2 + 6x - 3
6x^2 + 6x - 3 = 0
2x^2 + 2x - 1 = 0
(2x 2)(x 1) or (2x 1)(x 2). . .no, doesn't look like it factors nicely.
Quadratic
x = (-2 +- sqrt(2^2 - 4(2)(-1))) / 2(2)
x = (-2 +- sqrt(4 + 8)) / 4
x = (-2 +- sqrt(12)) / 4
x = (-2 +- 2sqrt(3)) / 4
x = (-1 +- sqrt(3)) / 2
Answer: x = (-1 + sqrt(3)) / 2, or x = (-1 - sqrt(3)) / 2