The image contains the following mathematical expression and instructions:

1. You are given:
   $(g \circ f)(x) = 3x^2 + 4$

2. It is also stated:
   $f(x) = x + 1$, and the task is to determine $g(x)$.

We are tasked with finding the function $g(x)$, given the composition $(g \circ f)(x)$ and $f(x)$.

Step-by-Step Solution:

1. The composition of functions $(g \circ f)(x)$ means $g(f(x))$. This implies that: $g(f(x)) = 3x^2 + 4$

2. From the given, we know that $f(x) = x + 1$. Substituting this into the expression for $g(f(x))$: $g(x + 1) = 3x^2 + 4$

3. Now, to find $g(x)$, let’s substitute $y = x + 1$, which gives $x = y - 1$. This transforms the equation into: $g(y) = 3(y - 1)^2 + 4$

4. Expand the expression on the right-hand side: $g(y) = 3(y^2 - 2y + 1) + 4$ Simplify it: $g(y) = 3y^2 - 6y + 3 + 4$ $g(y) = 3y^2 - 6y + 7$

Therefore, the function $g(x)$ is: $g(x) = 3x^2 - 6x + 7$

Let me know if you would like further details or clarifications!

Here are 5 related questions that expand on this topic:

1. How would the solution change if $f(x)$ were a different function, say $f(x) = 2x + 3$?
2. What is the general procedure for finding the composition of two functions?
3. Can you solve for $f(x)$ if $g(x)$ and $(g \circ f)(x)$ are known?
4. How does the composition of functions relate to the chain rule in calculus?
5. What is the geometric interpretation of composing two functions?

Tip: When solving compositions, focus on expressing one function entirely in terms of the other to simplify the process.