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To analyze the asymptotes of the given function 

\[
y = \frac{|x| \sqrt{9x^4 + 6x^2 + 2}}{(2x + 3)(x + 1)^2}
\]

we need to find both horizontal and vertical asymptotes. Let's break down each part.

### (a) Horizontal Asymptotes
To find horizontal asymptotes, we examine the behavior of \( y \) as \( x \to \infty \) and \( x \to -\infty \). 

1. **Numerator Analysis:**
   - The term inside the square root in the numerator is \(9x^4 + 6x^2 + 2\).
   - As \( x \to \infty \) (or \( x \to -\infty \)), the highest power term dominates, so \( \sqrt{9x^4 + 6x^2 + 2} \approx \sqrt{9x^4} = 3x^2 \).
   - Since the numerator has \( |x| \cdot 3x^2 \), it approximates to \( 3x^3 \) for large \( x \).

2. **Denominator Analysis:**
   - The denominator is \( (2x + 3)(x + 1)^2 \).
   - Expanding the highest degree terms, this becomes approximately \( 2x \cdot x^2 = 2x^3 \).

3. **Limit at Infinity:**
   - Now, consider the ratio \( \frac{3x^3}{2x^3} = \frac{3}{2} \).
   - Therefore, as \( x \to \infty \) and \( x \to -\infty \), \( y \to \frac{3}{2} \).
   
So, the horizontal asymptote is:
\[
y = \frac{3}{2}
\]

### (b) Vertical Asymptotes
To find vertical asymptotes, we need to identify where the function is undefined due to division by zero in the denominator. This occurs when:

1. \( 2x + 3 = 0 \Rightarrow x = -\frac{3}{2} \)
2. \( (x + 1)^2 = 0 \Rightarrow x = -1 \)

Since both \( x = -\frac{3}{2} \) and \( x = -1 \) make the denominator zero, they are potential vertical asymptotes. We confirm that neither point causes the numerator to also be zero, so these values are indeed vertical asymptotes.

Thus, the vertical asymptotes are:
\[
x = -\frac{3}{2} \quad \text{and} \quad x = -1
\]

### Summary
- **Horizontal Asymptote**: \( y = \frac{3}{2} \)
- **Vertical Asymptotes**: \( x = -\frac{3}{2} \) and \( x = -1 \)

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

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Here are 5 related questions that might deepen your understanding:

1. How do we determine horizontal asymptotes for rational functions in general?
2. What role does the absolute value in the numerator play in analyzing limits?
3. How would the presence of a higher degree in the numerator affect the asymptotes?
4. Why do vertical asymptotes occur where the denominator is zero?
5. Can a function have more than one horizontal asymptote?

**Tip:** When analyzing limits to find horizontal asymptotes, focus only on the terms with the highest power of \( x \) in both the numerator and denominator.

Find (a) all horizontal and (b) all vertical asymptotes of the graph

y = \frac{|x| \sqrt{9x^4 + 6x^2 + 2}}{(2x + 3)(x + 1)^2}

Learn how to determine the horizontal and vertical asymptotes of the rational function y = \frac{|x| \sqrt{9x^4 + 6x^2 + 2}}{(2x + 3)(x + 1)^2}. This problem explores the use of limits and asymptotic behavior, suitable for students in Grades 11-12.

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