To solve this equation, we will first simplify the expression on the left side:

(x+2)^(3) - (x+1)(x-2x+1) = 1

(x+2)(x+2)(x+2) - (x+1)(-x+1) = 1

(x+2)(x+2)(x+2) + (x+1)(x-1) = 1

(x+2)(x+2)(x+2) + (x^2 - 1) = 1

(x+2)(x+2)(x+2) + x^2 - 1 = 1

(x+2)(x+2)(x+2) + x^2 = 2

Now, let's expand the expression (x+2)(x+2)(x+2):

(x+2)(x+2)(x+2) + x^2 = 2

(x^2 + 4x + 4)(x+2) + x^2 = 2

(x^3 + 2x^2 + 4x^2 + 8x + 4x + 8) + x^2 = 2

x^3 + 6x^2 + 12x + 8 + x^2 = 2

x^3 + 7x^2 + 12x + 8 + x^2 = 2

x^3 + 8x^2 + 12x + 8 = 2

Now, let's combine like terms:

x^3 + 8x^2 + 12x + 8 - 2 = 0

x^3 + 8x^2 + 12x + 6 = 0

This is the simplified form of the equation.