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The given expression is:

\[
\frac{y^2 - 6y + 5}{y^2 + 3y + 2} \div \frac{y^2 - 25}{y^2 + 6y + 5}
\]

To divide these rational expressions, we will multiply the first fraction by the reciprocal of the second fraction.

This becomes:

\[
\frac{y^2 - 6y + 5}{y^2 + 3y + 2} \times \frac{y^2 + 6y + 5}{y^2 - 25}
\]

Now, let's factor each of the polynomials where possible:

1. \( y^2 - 6y + 5 = (y - 5)(y - 1) \)
2. \( y^2 + 3y + 2 = (y + 2)(y + 1) \)
3. \( y^2 + 6y + 5 = (y + 5)(y + 1) \)
4. \( y^2 - 25 = (y - 5)(y + 5) \)

Now substitute the factored forms into the expression:

\[
\frac{(y - 5)(y - 1)}{(y + 2)(y + 1)} \times \frac{(y + 5)(y + 1)}{(y - 5)(y + 5)}
\]

Next, cancel out the common factors:

- \( (y - 5) \) cancels with \( (y - 5) \)
- \( (y + 5) \) cancels with \( (y + 5) \)
- \( (y + 1) \) cancels with \( (y + 1) \)

After simplifying, we are left with:

\[
\frac{y - 1}{y + 2}
\]

Thus, the simplified expression is:

\[
\frac{y - 1}{y + 2}
\]

Would you like further explanation or details on this? Here are a few related questions:

1. How do you find common factors when simplifying rational expressions?
2. What are the steps to factor a quadratic trinomial like \( y^2 - 6y + 5 \)?
3. Can you explain the rules for dividing two rational expressions in more detail?
4. How does multiplying by the reciprocal help in dividing fractions?
5. How do you know when to cancel terms in rational expressions?

**Tip:** When dividing rational expressions, always check if factoring can simplify the terms before multiplying.

Divide the expression: (y^2 - 6y + 5)/(y^2 + 3y + 2) ÷ (y^2 - 25)/(y^2 + 6y + 5)

Learn how to divide and simplify rational expressions involving quadratics. The step-by-step solution covers factoring, multiplying by the reciprocal, and canceling common terms. Suitable for Grades 10-12 students.

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