The problem asks for the composition of functions $(f \circ g)(x)$, given:

$f(x) = 3x + 3 \quad \text{and} \quad g(x) = 2x^2 + 2x.$

The composition $(f \circ g)(x)$ means substituting $g(x)$ into $f(x)$. Here's the solution:

Step 1: Write the composition

$(f \circ g)(x) = f(g(x)) = f(2x^2 + 2x).$

Step 2: Substitute $g(x) = 2x^2 + 2x$ into $f(x) = 3x + 3$

$f(2x^2 + 2x) = 3(2x^2 + 2x) + 3.$

Step 3: Distribute and simplify

$3(2x^2 + 2x) + 3 = 6x^2 + 6x + 3.$

Final Answer:

$(f \circ g)(x) = 6x^2 + 6x + 3.$

Would you like further steps explained or additional examples?

Here are 5 questions to expand your understanding:

1. What is the composition $(g \circ f)(x)$ for the same functions?
2. How do you find the domain of $(f \circ g)(x)$?
3. What happens if $g(x)$ is a constant function instead of a polynomial?
4. Can you reverse the order $(f \circ g)$ and $(g \circ f)$? Why or why not?
5. How does composition work with trigonometric functions?

Tip: When composing functions, always remember to substitute the entire expression of one function into the variable of the other!