Let's evaluate the given algebraic expression for $x = -4$.

The expression is:

$-x^2 - 2$

Substitute $x = -4$ into the expression:

$-(-4)^2 - 2$

First, calculate $(-4)^2$:

$(-4)^2 = 16$

Now substitute $16$ back into the expression:

$-16 - 2$

Finally, simplify:

$-16 - 2 = -18$

So, the value of the expression when $x = -4$ is:

$\boxed{-18}$

Would you like any more details on this? Here are some follow-up questions you might find interesting:

1. What is the difference between evaluating $-x^2$ and $(-x)^2$?
2. How does the order of operations affect evaluating this expression?
3. What happens when you substitute a positive value for $x$ in this expression?
4. How would this expression change if you added another term, say $+ 3x$?
5. Can you graph the expression $-x^2 - 2$ and analyze its shape?

Tip: Always pay attention to whether the negative sign is inside or outside the parentheses when dealing with exponents. It makes a big difference!