1) Initial expression: -n^2+5n+24
2) Factoring out negative one: -1(n^2-5n-24)
3) Factorized form: -1(n-8)(n-8)
In the next step, what does happen with with the minus one if I am to solve for n?
Also, if in the second step -1 corresponds to the whole expression inside the parantheses, why do we factor out -1 just in front of the first (n-8)?
If you have to "solve" for a variable, say $n$, you have to set the quadratic equal to $0$ to find its roots.
$$-n^2+5n+24=0$$
Here, you can multiply both sides by $-1$:
$$-1\times(-n^2+5n+24)=-1\times(0)$$
$$n^2-5n-24=0$$
Which finally results in:
$$(n-8)(n+3)=0$$