We are asked to find $f(g(x))$ where:

• $f(x) = -3x - 10$
• $g(x) = 3x^2 - 6x - 12$

To find $f(g(x))$, we substitute $g(x)$ into $f(x)$. That is, we replace $x$ in $f(x)$ with $g(x)$.

Steps:

1. Substitute $g(x) = 3x^2 - 6x - 12$ into $f(x) = -3x - 10$. $f(g(x)) = -3(3x^2 - 6x - 12) - 10$

2. Distribute the $-3$ across the terms inside the parentheses: $f(g(x)) = -9x^2 + 18x + 36 - 10$

3. Simplify the constant terms: $f(g(x)) = -9x^2 + 18x + 26$

Thus, the function $f(g(x))$ is: $f(g(x)) = -9x^2 + 18x + 26$

Would you like further explanation or details on any step?

Here are 5 related questions you can explore:

1. What is $g(f(x))$ for the same functions?
2. How do you find the domain of $f(g(x))$?
3. What happens to the graph of $f(x)$ when composed with $g(x)$?
4. Can you express $f(g(x))$ in factored form?
5. How would the composition change if $g(x) = 3x^2 + 6x - 12$?

Tip: When composing functions, always replace the input variable with the entire second function carefully.