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Let's go through the problem step by step. The given problem is to divide two rational expressions:

\[
\frac{y^2 + 9y + 18}{y^2 + 11y + 30} \div \frac{y^2 - 10y + 16}{y^2 + 3y - 10}
\]

### Step 1: Rewrite the division as multiplication
When dividing fractions, you multiply by the reciprocal of the second fraction. So, we rewrite the expression as:

\[
\frac{y^2 + 9y + 18}{y^2 + 11y + 30} \times \frac{y^2 + 3y - 10}{y^2 - 10y + 16}
\]

### Step 2: Factor each polynomial

#### First numerator: \( y^2 + 9y + 18 \)
This factors as:
\[
y^2 + 9y + 18 = (y + 3)(y + 6)
\]

#### First denominator: \( y^2 + 11y + 30 \)
This factors as:
\[
y^2 + 11y + 30 = (y + 5)(y + 6)
\]

#### Second numerator: \( y^2 + 3y - 10 \)
This factors as:
\[
y^2 + 3y - 10 = (y + 5)(y - 2)
\]

#### Second denominator: \( y^2 - 10y + 16 \)
This factors as:
\[
y^2 - 10y + 16 = (y - 2)(y - 8)
\]

### Step 3: Substitute the factored forms
Now, substitute the factored polynomials into the expression:

\[
\frac{(y + 3)(y + 6)}{(y + 5)(y + 6)} \times \frac{(y + 5)(y - 2)}{(y - 2)(y - 8)}
\]

### Step 4: Cancel out common factors
- \( (y + 6) \) appears in both the numerator and denominator, so it can be canceled out.
- \( (y + 5) \) also appears in both the numerator and denominator, so it can be canceled out.
- \( (y - 2) \) appears in both the numerator and denominator, so it can be canceled out as well.

The remaining expression is:

\[
\frac{y + 3}{y - 8}
\]

### Final Answer:
\[
\frac{y + 3}{y - 8}
\]

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. How do you factor trinomials of the form \(ax^2 + bx + c\)?
2. What is the rule for dividing rational expressions?
3. Can you explain how to cancel out common factors in a rational expression?
4. How do you handle a situation where one of the factors is not common between numerator and denominator?
5. What are the conditions where division by a certain term is not allowed (undefined)?

**Tip:** Always double-check your factored forms to ensure no terms are missing or incorrect.

Divide as indicated: (y^2 + 9y + 18) / (y^2 + 11y + 30) ÷ (y^2 - 10y + 16) / (y^2 + 3y - 10).

Learn how to divide rational expressions such as (y^2 + 9y + 18) / (y^2 + 11y + 30) ÷ (y^2 - 10y + 16) / (y^2 + 3y - 10) by factoring quadratics and simplifying the result step-by-step. Suitable for Grade 8-10 students.

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