Let's expand the expressions and then simplify:

First, expand the expression (x+6)(x-2):

(x+6)(x-2) = x(x-2) + 6(x-2)
= x^2 - 2x + 6x - 12
= x^2 + 4x - 12

Next, expand the expression (x+1)(x-3):

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

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

(x^2 + 4x - 12) - (x^2 - 2x - 3) = 0
x^2 + 4x - 12 - x^2 + 2x + 3 = 0
6x - 9 = 0
6x = 9
x = 9/6
x = 1.5

Therefore, the solution to the equation is x = 1.5.