https://file.aiquickdraw.com/mathbot/file/1727362054858365ex344.jpg

The function given in the image is:

\[
r(x) = \frac{x^2 - x - 2}{(x + 1)^2(x - 2)}
\]

We are tasked with finding which statement about the holes in the graph of \( r(x) \) is correct.

### Step-by-Step Solution:

1. **Factor the numerator**:
   \[
   x^2 - x - 2 = (x - 2)(x + 1)
   \]
   So, the function can be rewritten as:
   \[
   r(x) = \frac{(x - 2)(x + 1)}{(x + 1)^2(x - 2)}
   \]

2. **Simplify the expression**:
   - The factor \( (x + 1) \) in the numerator can cancel with one of the \( (x + 1) \) terms in the denominator, so we have:
     \[
     r(x) = \frac{1}{x + 1}, \quad \text{with restrictions}.
     \]
   - However, there are restrictions on the domain because of the original factors in the denominator. Specifically:
     - At \( x = -1 \), the original denominator becomes zero due to the \( (x + 1)^2 \) term, which would cause a hole in the graph.
     - At \( x = 2 \), both the numerator and denominator have a factor of \( (x - 2) \), which indicates another hole at \( x = 2 \).

3. **Multiplicity Analysis**:
   - For \( x = -1 \), the factor \( (x + 1) \) appears twice in the denominator (with a power of 2) and once in the numerator. This means the multiplicity of \( x = -1 \) is greater in the denominator, leading to a hole.
   - For \( x = 2 \), the factor \( (x - 2) \) appears both in the numerator and denominator with equal multiplicities, indicating a hole as well.

### Conclusion:
There are holes at both \( x = -1 \) and \( x = 2 \) because:
- The multiplicity of \( x = -1 \) in the denominator is greater than in the numerator.
- The multiplicity of \( x = 2 \) in the numerator and denominator is equal.

Thus, the correct answer is **Option A**.

---

Would you like a detailed explanation of any of these steps? Here are some related questions for deeper understanding:

1. How do you find holes in the graph of a rational function?
2. What is the significance of multiplicity in determining the nature of a hole?
3. How do you simplify rational expressions involving polynomials?
4. What are the conditions for a vertical asymptote versus a hole in a graph?
5. How do repeated factors in the denominator affect the graph of a rational function?

**Tip**: Always check for holes by simplifying the function and examining common factors in the numerator and denominator before and after simplification.

The function r is given by r(x) = (x^2 - x - 2) / ((x + 1)^2(x - 2)). In the xy-plane, which of the following is true about holes in the graph of r?

Explore how to identify holes in the graph of a rational function given by r(x) = (x^2 - x - 2) / ((x + 1)^2(x - 2)). Learn about factoring, simplifying expressions, and understanding multiplicity to determine the presence of holes at specific points. Suitable for students in Grades 10-12.

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