To solve this equation, we first need to isolate the square roots on one side of the equation and square both sides to get rid of the square roots.

√(x+3) - √(7-x) = √(2x-8)

First, let's move one of the square roots to the other side of the equation:

√(x+3) = √(7-x) + √(2x-8)

Now, square both sides of the equation to eliminate the square roots:

(x+3) = (7-x) + 2√[(7-x)(2x-8)] + (2x-8)

Expanding the terms gives:

x + 3 = 7 - x + 2√[14x - 2x^2 - 16x + 64] + 2x - 8

Combining like terms and simplifying further:

x + 3 = 7 - x + 2√[14x - 2x^2 - 16x + 64] + 2x - 8
x + 3 = -x + 6 + 2√[-2x^2 - 2x + 64]
2x = 3 + 2√[64 - 2(x^2 + x)]

Now square both sides again to eliminate the square root:

(2x)^2 = (3 + 2√[64 - 2(x^2 + x)])^2
4x^2 = 9 + 12√[64 - 2(x^2 + x)] + 4(64 - 2(x^2 + x))
4x^2 = 9 + 12√[64 - 2(x^2 + x)] + 256 - 8(x^2 + x)

Rearranging and simplifying gives:

4x^2 = 265 - 8x^2 - 4x + 12√[64 - 2(x^2 + x)]
12x^2 + 4x - 265 = 12√[64 - 2(x^2 + x)]

To continue solving for x, we would need to isolate the square root term and then square both sides once more. This process may result in a complex solution, so it's recommended to check for any extraneous solutions.