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To solve the given problem, we are asked to divide the following rational expressions:

\[
\frac{y^2 - 25}{y - 5} \div \frac{y + 5}{2y - 10}
\]

### Step 1: Rewrite Division as Multiplication
When dividing fractions, we multiply the first fraction by the reciprocal of the second. So, the expression becomes:

\[
\frac{y^2 - 25}{y - 5} \times \frac{2y - 10}{y + 5}
\]

### Step 2: Factor the Numerators and Denominators
Now, let's factor both the numerator and the denominator wherever possible.

1. \( y^2 - 25 \) is a difference of squares, so it can be factored as:
   \[
   y^2 - 25 = (y - 5)(y + 5)
   \]

2. \( 2y - 10 \) can be factored by factoring out the greatest common factor:
   \[
   2y - 10 = 2(y - 5)
   \]

Now, rewrite the expression with the factored forms:
\[
\frac{(y - 5)(y + 5)}{y - 5} \times \frac{2(y - 5)}{y + 5}
\]

### Step 3: Cancel Common Factors
Now, cancel out the common factors in the numerator and denominator:
- \( y - 5 \) appears in both the numerator and denominator, so we cancel it.
- \( y + 5 \) also appears in both the numerator and denominator, so we cancel it as well.

The simplified expression becomes:
\[
2
\]

### Step 4: Determine the Excluded Values
We also need to exclude any values that would make the original denominators equal to zero.

1. From the original denominator \( y - 5 \), we get \( y \neq 5 \).
2. From the second denominator \( 2y - 10 = 0 \), we solve for \( y \):
   \[
   2y - 10 = 0 \implies y = 5
   \]
   So, \( y \neq 5 \) (already covered).
3. Finally, from the denominator \( y + 5 \), we get \( y \neq -5 \).

Thus, the excluded values are \( y \neq 5, -5 \).

### Step 5: Conclusion
The correct simplified expression is \( 2 \), and the excluded values are \( y \neq 5, -5 \).

The correct answer is **option A**, and the box should be filled with \( 2 \).

---

Do you have any questions or would you like more details? 

Here are 5 related questions to deepen your understanding:
1. How do you factor the difference of squares?
2. What is the process of finding the reciprocal of a fraction?
3. How do you identify values to exclude in a rational expression?
4. Why is canceling common factors valid in algebraic expressions?
5. Can you explain when it is not possible to simplify a rational expression further?

**Tip**: Always double-check excluded values, as they are critical to ensuring the domain of the function remains valid.

Divide the following rational expression: (y^2 - 25) / (y - 5) ÷ (y + 5) / (2y - 10). Select the correct choice and simplify the expression.

Learn how to divide and simplify rational expressions using algebraic techniques. This problem demonstrates the use of the difference of squares and factoring concepts to simplify the expression. It is suitable for students in Grades 9-12.

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