We are given two functions:

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

We are asked to find $f(g(x))$, which is the composition of the functions $f$ and $g$.

Step 1: Substitute $g(x)$ into $f(x)$

To find $f(g(x))$, we substitute the expression for $g(x)$ into the function $f$. This means we replace every occurrence of $x$ in $f(x)$ with $g(x)$.

$f(g(x)) = -3(g(x)) - 10$

Substitute $g(x) = 3x^2 - 6x - 12$ into this expression:

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

Step 2: Simplify

Now distribute the $-3$ to each term inside the parentheses:

$f(g(x)) = -9x^2 + 18x + 36 - 10$

Combine the constants:

$f(g(x)) = -9x^2 + 18x + 26$

So, the composition $f(g(x))$ is:

$f(g(x)) = -9x^2 + 18x + 26$

Final Answer:

$f(g(x)) = -9x^2 + 18x + 26$

Would you like further details or explanations?

Here are five follow-up questions you might consider:

1. How do we graph the composition $f(g(x))$?
2. What is the domain of $f(g(x))$?
3. Can we find the inverse of the composed function $f(g(x))$?
4. How would the result change if $f(x)$ were a quadratic function?
5. What are the key points where $f(g(x)) = 0$?

Tip: Always substitute carefully and simplify step-by-step to avoid errors in function composition.