Let's solve the given problem step-by-step.

We are given:

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

We are tasked with finding $(f \circ g)(x)$, which means $f(g(x))$.

Steps:

1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = f(2x^2 + 2x)$.

2. Replace $x$ in $f(x)$ with $g(x)$: Since $f(x) = 3x + 3$, replacing $x$ with $g(x)$ gives: $f(g(x)) = 3(2x^2 + 2x) + 3$

3. Simplify the expression: Distribute the $3$ across the terms inside the parentheses: $f(g(x)) = 6x^2 + 6x + 3$

Final Answer:

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

Would you like further explanation, or should I clarify any part of the solution?

Here are 5 related questions you might find interesting:

1. How do we find $(g \circ f)(x)$, and how does it differ from $(f \circ g)(x)$?
2. What are the domain and range of the composite function $(f \circ g)(x)$?
3. How do we compute the derivative of $(f \circ g)(x)$ using the chain rule?
4. What happens if $g(x)$ is a trigonometric function instead of a polynomial?
5. How do we verify that a given function is a valid composition of two functions?

Tip: Always substitute carefully when dealing with composite functions to avoid errors.