Expanding both sides of the equation:

(x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6

(x-2)(x-4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8

Setting the two expressions equal to each other:

x^2 + 5x + 6 = x^2 - 6x + 8

Subtracting x^2 from both sides:

5x + 6 = -6x + 8

Adding 6x to both sides:

11x + 6 = 8

Subtracting 6 from both sides:

11x = 2

Dividing by 11:

x = 2/11

Therefore, x = 2/11 is the solution to the given equation.