The problem asks to find $(f \circ g)(x)$, which represents the composition of the functions $f(x)$ and $g(x)$. This means you need to substitute $g(x)$ into $f(x)$.

Given: $f(x) = 4x + 4 \quad \text{and} \quad g(x) = 4x^2 + 4x,$ we compute $(f \circ g)(x)$ as follows:

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

$f(g(x)) = f(4x^2 + 4x).$

Step 2: Replace every instance of $x$ in $f(x) = 4x + 4$ with $g(x) = 4x^2 + 4x$

$f(g(x)) = 4(4x^2 + 4x) + 4.$

Step 3: Simplify the expression

$f(g(x)) = 16x^2 + 16x + 4.$

Thus, the simplified result is: $(f \circ g)(x) = 16x^2 + 16x + 4.$

Would you like further clarification on any of these steps? Here are 5 related questions to expand your understanding:

1. How do you compute the composition of two functions in general?
2. Can $(f \circ g)(x)$ always be simplified, or are there cases where it’s more complex?
3. What is the difference between $f(g(x))$ and $g(f(x))$?
4. How does function composition differ from simple function addition or multiplication?
5. Could you compute the derivative of $(f \circ g)(x)$ using the chain rule?

Tip: When performing function composition, always ensure each function's range can correctly map into the domain of the next function for accurate results.