To simplify this expression, let's expand each term:

(x+1)(x+3) = x^2 + 4x + 3
(x+3)(x+2) = x^2 + 5x + 6
(x+2)(x+4) = x^2 + 6x + 8

Now, substitute these expanded expressions back into the original equation:

(x^2 + 4x + 3) - (x^2 + 5x + 6) + (x^2 + 6x + 8) = 0

Now, simplify this equation by combining like terms:

x^2 + 4x + 3 - x^2 - 5x - 6 + x^2 + 6x + 8 = 0

4x - 5x + 6x = 5x
3 - 6 + 8 = 5

Therefore, the simplified equation is:
5x + 5 = 0

Now solve for x:
5x = -5
x = -1

So, the solution to the equation (x+1)(x+3) - (x+3)(x+2) + (x+2)(x+4) = 0 is x = -1.