The problem seems to involve the functions $f(x) = x^2 + 5x + 6$ and $g(x) = 2x + 4$, and it asks to identify or relate them to the given expressions.

Let's break it down:

1. Factorization of $f(x)$

Given: $f(x) = x^2 + 5x + 6$

This quadratic expression can be factored as: $f(x) = (x + 3)(x + 2)$

2. Simplification of $g(x)$

Given: $g(x) = 2x + 4$

This linear expression can be factored as: $g(x) = 2(x + 2)$

Comparison with Given Options:

A. $\frac{(x + 3)(x + 2)}{2x + 4}$:

Simplifying the denominator: $\frac{(x + 3)(x + 2)}{2(x + 2)}$ The $(x + 2)$ terms cancel out: $\frac{x + 3}{2}$

This matches option C.

B. $(x + 3)(x + 2)$:

This is exactly $f(x)$, so this is the factorization of $f(x)$.

C. $\frac{x + 3}{2}$:

This is the simplified result from A.

D. $\frac{2x + 4}{x + 3}$:

Simplifying the numerator: $\frac{2(x + 2)}{x + 3}$ This doesn't simplify to anything from the previous expressions.

Conclusion:

• Option A simplifies to Option C.
• Option B is the factorization of $f(x)$.
• Option D is the fraction of $g(x)$ over $x + 3$, which doesn't directly match the given options when simplified.

Would you like more details on any part of this process, or do you have any questions?

Here are 5 related questions to explore:

1. How do you factor a quadratic equation like $f(x)$?
2. What are the steps to simplify rational expressions?
3. How do you solve for $x$ if given $f(x) = g(x)$?
4. What happens if you substitute specific values of $x$ into these functions?
5. How does the graph of $f(x) = x^2 + 5x + 6$ compare to the graph of $g(x) = 2x + 4$?

Tip: Always check for common factors in both the numerator and denominator before simplifying rational expressions.