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The expression you've provided is a multiplication of two rational expressions. Let's solve it step by step.

### The Expression:
\[
\frac{2a^2 + 7a + 6}{2a^2 + 9a + 9} \times \frac{2a^2 + 17a + 8}{4a^2 + 9a + 2}
\]

### Step 1: Factor the Numerators and Denominators
First, we'll factor each quadratic expression if possible.

1. **Factor \(2a^2 + 7a + 6\):**
   - To factor \(2a^2 + 7a + 6\), we look for factors of \(2 \times 6 = 12\) that add up to 7.
   - The factors are 3 and 4.
   - So, \(2a^2 + 7a + 6 = (2a + 3)(a + 2)\).

2. **Factor \(2a^2 + 9a + 9\):**
   - This quadratic can be factored as \(2a^2 + 9a + 9 = (2a + 3)(a + 3)\).

3. **Factor \(2a^2 + 17a + 8\):**
   - We look for factors of \(2 \times 8 = 16\) that add up to 17.
   - The factors are 16 and 1.
   - So, \(2a^2 + 17a + 8 = (2a + 1)(a + 8)\).

4. **Factor \(4a^2 + 9a + 2\):**
   - We need factors of \(4 \times 2 = 8\) that add up to 9.
   - The factors are 8 and 1.
   - So, \(4a^2 + 9a + 2 = (4a + 1)(a + 2)\).

### Step 2: Substitute the Factored Forms
Now, we can rewrite the expression with the factored forms:

\[
\frac{(2a + 3)(a + 2)}{(2a + 3)(a + 3)} \times \frac{(2a + 1)(a + 8)}{(4a + 1)(a + 2)}
\]

### Step 3: Simplify by Canceling Common Factors
- Notice that \((2a + 3)\) in the numerator and denominator of the first fraction can be canceled.
- Also, \((a + 2)\) in the numerator of the first fraction and the denominator of the second fraction can be canceled.

After canceling, the expression simplifies to:

\[
\frac{1 \times (2a + 1)(a + 8)}{(a + 3)(4a + 1)}
\]

### Final Simplified Expression:
\[
\frac{(2a + 1)(a + 8)}{(a + 3)(4a + 1)}
\]

Would you like to go over any of these steps in more detail or ask further questions?

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### 5 Related Questions:
1. How do you factor quadratic expressions in general?
2. What are common strategies for simplifying rational expressions?
3. How do you identify which terms to cancel when simplifying a rational expression?
4. Can all quadratic expressions be factored? If not, how do you solve them?
5. How can you apply this technique to solve equations involving rational expressions?

### Tip:
Always check if any factors can be simplified before performing the multiplication or division of rational expressions. This can save time and reduce the complexity of the problem.

Solve the multiplication of two rational expressions: (2a^2 + 7a + 6) / (2a^2 + 9a + 9) * (2a^2 + 17a + 8) / (4a^2 + 9a + 2)

Learn how to simplify rational expressions by factoring quadratics and canceling common factors. This example solves the expression (2a^2 + 7a + 6) / (2a^2 + 9a + 9) * (2a^2 + 17a + 8) / (4a^2 + 9a + 2), showing step-by-step factoring and simplification.

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