To simplify this expression, let's first expand the terms in the expression:

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

Now, substitute the expanded terms into the equation:

[(x^2 + 2x + 1)/6] + [(x^2 - 2x + 1)/12] - x^2 - 1/4 = 1

Simplify the fractions by finding a common denominator of 12:

[(2(x^2 + 2x + 1) + (x^2 - 2x + 1) - 6x^2 - 3)/12] = 1

Now, combine like terms:

[(2x^2 + 4x + 2 + x^2 - 2x + 1 - 6x^2 - 3)/12] = 1
[(3x^2 + 2x)/12] = 1

Simplify the expression:

(3x^2 + 2x)/12 = 1
3x^2 + 2x = 12
3x^2 + 2x - 12 = 0

This is the simplified form of the original expression.