Let's solve the given problem step by step.

We are given two functions:

1. $f(x) = x^2 + x - 6$
2. $g(x) = \frac{x - 2}{x + 3}$

We are tasked with finding $f(g(2))$.

Step 1: Compute $g(2)$

Using the definition of $g(x)$, substitute $x = 2$:

$g(2) = \frac{2 - 2}{2 + 3} = \frac{0}{5} = 0$

So, $g(2) = 0$.

Step 2: Compute $f(g(2)) = f(0)$

Now that we know $g(2) = 0$, we substitute $x = 0$ into the function $f(x)$:

$f(0) = 0^2 + 0 - 6 = -6$

Therefore, $f(g(2)) = -6$.

Final Answer:

The value of $f(g(2))$ is $-6$, so the correct option is $-6$.

Would you like more details on any part of this solution?

Here are 5 related questions to expand your understanding:

1. What happens if $g(x)$ has a denominator that can cause division by zero?
2. Can we generalize this process for different functions $f(x)$ and $g(x)$?
3. What is the domain of the composite function $f(g(x))$?
4. How do we differentiate $f(g(x))$ if both functions are differentiable?
5. What is the importance of the order in which we apply functions in composition?

Tip: When working with function compositions, always check the domain of each function to avoid undefined values.