To find $f(g(-2))$, we need to follow these steps:

1. Evaluate $g(-2)$ first using the function $g(x) = x - x^2$.
2. Then, substitute the result into $f(x)$.

Step 1: Find $g(-2)$

We have the function $g(x) = x - x^2$.

Substitute $x = -2$ into $g(x)$: $g(-2) = (-2) - (-2)^2$ $g(-2) = -2 - 4 = -6$

Step 2: Find $f(g(-2)) = f(-6)$

Now that we know $g(-2) = -6$, we substitute $x = -6$ into the function $f(x) = 4x + 2$: $f(-6) = 4(-6) + 2$ $f(-6) = -24 + 2 = -22$

Thus, $f(g(-2)) = -22$.

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Would you like a more detailed breakdown or explanation? Here are some related questions you might find interesting:

1. How do composition functions work in general?
2. What is the general method for solving composite functions?
3. How do you evaluate composite functions involving quadratic expressions?
4. What happens when the functions $f(x)$ and $g(x)$ are inverses of each other?
5. How would the result change if $f(x) = 2x + 3$ instead of $4x + 2$?

Tip: Always start by evaluating the innermost function when working with composite functions, as the result of one feeds directly into the next function.