(a-2)(a+4)-(a+1)

To simplify the expression \((a-2)(a+4)-(a+1)\), we'll follow these steps:

1. Expand the first part, \((a - 2)(a + 4)\).
2. Simplify the expression by subtracting \((a + 1)\).

Let's start with the first part:

\[ (a - 2)(a + 4) = a^2 + 4a - 2a - 8 = a^2 + 2a - 8 \]

Now, we can substitute this back into the expression:

\[ a^2 + 2a - 8 - (a + 1) \]

Next, distribute the minus sign:

\[ a^2 + 2a - 8 - a - 1 \]

Now combine like terms:

\[ a^2 + (2a - a) + (-8 - 1) = a^2 + a - 9 \]

Thus, the simplified expression is:

\[ \boxed{a^2 + a - 9} \]