To solve this equation, we can first rewrite it as:

(5^-x)^2 - 30 * 5^-x + 125 = 0

Now, let's make a substitution to simplify the equation. Let y = 5^-x.

Now the equation becomes:

y^2 - 30y + 125 = 0

This is a quadratic equation that we can solve by factoring:

(y - 25)(y - 5) = 0

So, y = 25 or y = 5.

Now, let's substitute back y = 5^-x:

5^-x = 25
x = log5(25)
x = 2

Or,

5^-x = 5
x = log5(5)
x = 1

Therefore, the solutions to the equation 5^-2x - 30 * 5^-x + 125 = 0 are x = 2 and x = 1.