To simplify the given expression, we will first expand the two sets of parentheses using the distributive property.

First, we expand (a + 4)(a - 2):
(a + 4)(a - 2) = a(a) + a(-2) + 4(a) + 4(-2)
= a^2 - 2a + 4a - 8
= a^2 + 2a - 8

Next, we expand (2x + 8)(x + 1):
(2x + 8)(x + 1) = 2x(x) + 2x(1) + 8(x) + 8(1)
= 2x^2 + 2x + 8x + 8
= 2x^2 + 10x + 8

Therefore, the given expression -(a + 4)(a - 2) - (2x + 8)(x + 1) simplifies to:
-(a^2 + 2a - 8) - (2x^2 + 10x + 8)
= -a^2 - 2a + 8 - 2x^2 - 10x - 8
= -a^2 - 2a - 2x^2 - 10x

The simplified expression is -a^2 - 2a - 2x^2 - 10x.