To solve this inequality, first expand the terms on both sides of the equation:

(x+4)(x-2)-(x+5)(x+3) = x^2 - 2x + 4x - 8 - (x^2 + 3x + 5x + 15)

Now simplify the equation:

= x^2 - 2x + 4x - 8 - x^2 - 3x - 5x - 15
= 2x - 23

Now compare this expression to the right side of the original inequality:

2x - 23 ≤ -8x

Add 8x to both sides:

10x - 23 ≤ 0

Add 23 to both sides:

10x ≤ 23

Divide by 10 on both sides to find the final solution:

x ≤ 2.3

Therefore, the solution to the original inequality is x ≤ 2.3.