To solve the given equation:

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

We need to find a common denominator for all fractions on both sides. The common denominator of (x-1), (x+1), (x+2), and (x-2) is (x+1)(x-1)(x+2)(x-2) = (x^2-1)(x^2-4) = (x^4 - 5x^2 + 4)

Now, multiplying each fraction by the necessary factors to get the common denominator, we get:

[(x-3)(x+1)(x+2)(x-2) + (x+3)(x-1)(x+2)(x-2)] / (x^4 - 5x^2 + 4) = [(x+6)(x-1)(x+1)(x-2) + (x-6)(x-1)(x+1)(x+2)] / (x^4 - 5x^2 + 4)

Simplify both sides of the equation:

[(x^2 - 2)(x^2 - 4) + (x^2 + 4)(x^2 - 1)] / (x^4 - 5x^2 + 4) = [(x^2 + 4)(x^2 - 1) + (x^2 - 4)(x^2 - 1)] / (x^4 - 5x^2 + 4)

Expand and simplify:

[(x^4 -6x^2 + 8) + (x^4 + 3x^2 - 4)] / (x^4 - 5x^2 + 4) = [(x^4 + 3x^2 - 4) + (x^4 - 5x^2 + 4)] / (x^4 - 5x^2 + 4)

Combine like terms:

[2x^4 - 3x^2 + 4] / (x^4 - 5x^2 + 4) = [2x^4 - 2x^2] / (x^4 - 5x^2 + 4)

Now we deduce that for the equation to be valid, the two sides must be equal:

Therefore,

2x^4 - 3x^2 + 4 = 2x^4 - 2x^2

Subtract 2x^4 from both sides to eliminate them:

-3x^2 + 4 = -2x^2

Subtract -2x^2 from both sides:

-3x^2 + 2x^2 + 4 = 0

-x^2 + 4 = 0

-x^2 = -4

x^2 = 4

x = ±2

Therefore, the value of x is 2 or -2.