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The function provided is:

\[
f(x) = \frac{x^2 + 6x + 5}{x^2 + 4x + 3}
\]

This is a rational function, which can be analyzed in terms of its domain, asymptotes, and simplification if possible.

### Steps to Analyze:

1. **Domain**: The domain of a rational function is all real numbers except where the denominator is zero. 
   To find the domain, set the denominator equal to zero and solve for \(x\).

   \[
   x^2 + 4x + 3 = 0
   \]

   Factoring the denominator:

   \[
   (x + 1)(x + 3) = 0
   \]

   So, \(x = -1\) and \(x = -3\) are the values where the denominator is zero. Therefore, the domain is all real numbers except \(x = -1\) and \(x = -3\).

2. **Simplification**: Let's see if the numerator and the denominator have any common factors that would simplify the function.

   The numerator is \(x^2 + 6x + 5\), which can be factored:

   \[
   (x + 1)(x + 5)
   \]

   Now the function becomes:

   \[
   f(x) = \frac{(x + 1)(x + 5)}{(x + 1)(x + 3)}
   \]

   We can cancel the common factor \( (x + 1) \), but we must note that \(x = -1\) still cannot be included in the domain.

   Therefore, the simplified form is:

   \[
   f(x) = \frac{x + 5}{x + 3} \quad \text{for} \quad x \neq -1
   \]

3. **Vertical Asymptotes**: Vertical asymptotes occur where the denominator is zero after simplification, and the function does not cancel out. In this case, there is a vertical asymptote at \(x = -3\).

4. **Horizontal Asymptotes**: For rational functions, if the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients. Here, both the numerator and denominator are quadratic, and the leading coefficients are both 1. Hence, the horizontal asymptote is:

   \[
   y = 1
   \]

### Summary:
- **Domain**: \( x \in \mathbb{R}, \, x \neq -1, x \neq -3 \)
- **Simplified Function**: \( f(x) = \frac{x + 5}{x + 3} \)
- **Vertical Asymptote**: \( x = -3 \)
- **Horizontal Asymptote**: \( y = 1 \)

Do you need further details or clarifications?

### Related Questions:
1. How do you find the domain of a more complex rational function?
2. What happens at points where a factor cancels in a rational function?
3. Can you find slant asymptotes in functions with unequal degrees of the numerator and denominator?
4. How do you determine the behavior of a function near its asymptotes?
5. What is the significance of horizontal asymptotes in long-term behavior?

### Tip:
Always remember to check for common factors in rational functions before determining the asymptotes! Simplifying early can make later steps much easier.

Explore how to simplify the rational function (x^2 + 6x + 5) / (x^2 + 4x + 3) by factoring both the numerator and the denominator, finding the domain, and identifying vertical and horizontal asymptotes. This problem is suited for students in Grades 9-12.

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