We are given the function $f(x) = -x^2 + x - 9$ and asked to compute $f(-2) + f(4)$.

Step 1: Calculate $f(-2)$

Substitute $x = -2$ into the function $f(x)$:

$f(-2) = -(-2)^2 + (-2) - 9$

$f(-2) = -(4) - 2 - 9 = -4 - 2 - 9 = -15$

Step 2: Calculate $f(4)$

Substitute $x = 4$ into the function $f(x)$:

$f(4) = -(4)^2 + 4 - 9$

$f(4) = -(16) + 4 - 9 = -16 + 4 - 9 = -21$

Step 3: Add $f(-2)$ and $f(4)$

Now, add the two results together:

$f(-2) + f(4) = -15 + (-21) = -36$

Thus, $f(-2) + f(4) = -36$.

Would you like further details or clarifications on any step?

Here are some related questions:

1. How do we find the roots of the quadratic function $f(x) = -x^2 + x - 9$?
2. What is the vertex of the function $f(x) = -x^2 + x - 9$?
3. How would you sketch the graph of $f(x) = -x^2 + x - 9$?
4. What is the maximum or minimum value of $f(x) = -x^2 + x - 9$?
5. How does changing the coefficient of $x^2$ affect the graph of the function?

Tip: The vertex form of a quadratic function can give you insights into the direction of the parabola and its maximum or minimum value.