To solve this equation, we need to simplify both sides and then find the common denominator.

Starting with the left side of the equation:
x/(2x-6) - x/(2x+2)

To find the common denominator, we multiply the first fraction by (2x+2)/(2x+2) and the second fraction by (2x-6)/(2x-6):

(x * (2x+2))/((2x-6)(2x+2)) - (x * (2x-6))/((2x+2)(2x-6))

Expanding the numerators, we get:

(2x^2 + 2x)/((2x-6)(2x+2)) - (2x^2 - 6x)/((2x+2)(2x-6))

Now, let's find the common denominator:

(2x^2 + 2x - (2x^2 - 6x))/((2x-6)(2x+2))

Simplifying the numerator:

(2x^2 + 2x - 2x^2 + 6x)/((2x-6)(2x+2))

Combining like terms:

(8x)/((2x-6)(2x+2))

Now, let's simplify the right side of the equation:

3x+2/(x^2 - 2x - 3)

To factor the denominator, we need to find two numbers that multiply to -3 and add up to -2. The numbers are -3 and 1:

3x+2/((x-3)(x+1))

Now, we can rewrite the equation:

(8x)/((2x-6)(2x+2)) = 3x+2/((x-3)(x+1))

To eliminate the denominators, we can multiply both sides of the equation by ((2x-6)(2x+2)) * ((x-3)(x+1)):

((8x)/((2x-6)(2x+2))) * ((2x-6)(2x+2)) * ((x-3)(x+1)) = (3x+2/((x-3)(x+1))) * ((2x-6)(2x+2)) * ((x-3)(x+1))

Simplifying both sides:

8x * ((x-3)(x+1)) = (3x+2) * ((2x-6)(2x+2))

Expanding both sides:

8x * (x^2 - 2x - 3) = (3x+2) * (4x^2 - 8x - 12)

Multiplying:

8x^3 - 16x^2 - 24x = 12x^3 - 24x^2 - 36x + 8x^2 - 16x - 24

Combining like terms:

8x^3 - 16x^2 - 24x = 12x^3 - 16x^2 - 52x - 24

Moving all terms to one side of the equation:

8x^3 - 16x^2 - 24x - 12x^3 + 16x^2 + 52x + 24 = 0

Simplifying:

-4x^3 + 28x + 24 = 0

This equation does not have a simple solution. It may require numerical methods or approximations to find the values of x that satisfy the equation.