10. Given f(x) = x ^ 2 + 5x + 6 and g(x) = 2x + 4 what

A. ((x + 3)(x + 2))/(2x + 4)

C. (x + 3)/2

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

D. (2x + 4)/(x + 3)

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.

Explore the factorization of quadratic equations and simplification of linear functions with examples of f(x) = x^2 + 5x + 6 and g(x) = 2x + 4. This detailed solution covers key algebraic concepts and formulas, suitable for Grades 10-12.

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