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The image shows a problem asking to factor an expression and identify one of the factors. The expression is:

\[
\frac{n+3}{x} + \frac{n+2}{x^2} + \frac{n+1}{y} + \frac{n}{y^2}
\]

We are asked to factor this expression and choose one of the given options:

A. \(x + y\)  
B. \(x^2 + y\)  
C. \(x^3 + y\)  
D. \(x^2 + y^2\)

Let's begin by simplifying the expression step by step.

1. Combine terms grouped by common factors. The terms are split into two groups based on \(x\) and \(y\):
   - \( \frac{n+3}{x} + \frac{n+2}{x^2} \)
   - \( \frac{n+1}{y} + \frac{n}{y^2} \)

2. Factoring each group:
   - The first group: \(\frac{n+3}{x} + \frac{n+2}{x^2} = \frac{(n+2)x + (n+3)}{x^2}\)
   - The second group: \(\frac{n+1}{y} + \frac{n}{y^2} = \frac{n(y+1)}{y^2}\)

Now, we can analyze the options, but to fully factor this expression and match it to the given answers, we might need to use a common factorization pattern. However, based on inspection, **B. \(x^2 + y\)** seems the closest to matching the structure of the factored terms.

Would you like further details on the steps, or do you have any questions? Here are 5 related questions you might find useful:

1. How do you approach factoring expressions with multiple variables?
2. What common factorization techniques work for rational expressions?
3. Can the given expression be factored differently using advanced methods?
4. How does grouping terms help in simplifying complex expressions?
5. What are some typical errors to avoid when factoring?

**Tip:** Always look for common factors first when trying to simplify or factor expressions.

Factoriza: \(\frac{n+3}{x} + \frac{n+2}{x^2} + \frac{n+1}{y} + \frac{n}{y^2}\). Señala uno de los factores.

Learn how to factor a complex rational expression involving variables raised to different powers. This problem explores factoring by grouping and analyzing terms with common factors. Suitable for students in Grades 10-12.

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