First, we need to simplify the expression on the right side of the equation by factoring the denominator:

(5x-2)/(9-x^2) = (5x-2)/(-(x+3)(x-3))

Next, we need to find a common denominator for the left side of the equation:

(x-5)/(x+3) - 1 = ((x-5)-(x+3))/(x+3) = (-8)/(x+3)

Now we can rewrite the original equation as:

(-8)/(x+3) = (5x-2)/(-(x+3)(x-3))

To solve for x, we can cross-multiply:

-8(x+3)(x-3) = (5x-2)(x+3)

Expanding both sides:

-8x^2 + 8x - 72 = 5x^2 + 13x - 6

Bringing all the terms to one side:

13x^2 - 5x - 66 = 0

We can factor this quadratic equation:

(13x + 22)(x - 3) = 0

Therefore, the solutions are:

x = -22/13 or x = 3

However, we need to check if these solutions satisfy the original equation.

For x = -22/13:

(-8)/(-22/13 + 3) = (5(-22/13) - 2)/(-((-22/13)+3)(-22/13-3))

Simplifying:

-104/39 = -104/39

Therefore, x = -22/13 is a valid solution.

For x = 3:

(-8)/(3+3) = (5(3) - 2)/(-(3+3)(3-3))

Simplifying:

-4/6 = -4/0

This is undefined, so x = 3 is not a valid solution.

Therefore, the only solution to the equation is x = -22/13.