To solve this inequality, we can simplify both sides by using the properties of exponents.

Simplify the left side of the inequality:
2^(x+3) - 2^(x+4) - 2^(x+5)
= 2^x 2^3 - 2^x 2^4 - 2^x 2^5
= 8 2^x - 16 2^x - 32 2^x
= -40 * 2^x

Simplify the right side of the inequality:
5^(x+2) - 5^(x+3)
= 5^x 5^2 - 5^x 5^3
= 25 5^x - 125 5^x
= -100 * 5^x

Now our inequality becomes:
-40 2^x <= -100 5^x

Divide both sides by 2^x to simplify:
-40 <= -100 * (5/2)^x

Now we have a new inequality in terms of (5/2)^x:
-40 <= -100 * (5/2)^x

To solve for x, we need to isolate (5/2)^x:
(5/2)^x >= 2/5

Since (5/2)^x is always positive, we can drop the absolute value symbols. Now we simply need to solve for x:
x >= log(base(5/2))(2/5)

Therefore, the solution to the inequality is x greater than or equal to log(base(5/2))(2/5).