Let's solve the given problem step by step.

Problem: Simplify the expression:

$\frac{x^2 + x - 6}{x^2 - 2x - 3} \cdot \frac{2x^2 - 3x - 9}{x^2 - x - 2} \div \frac{10x^2 + 27x + 18}{x^2 + 2x + 1}$

Step 1: Factorize all polynomials.

Numerator 1: $x^2 + x - 6$

Factorize: $x^2 + x - 6 = (x + 3)(x - 2)$

Denominator 1: $x^2 - 2x - 3$

Factorize: $x^2 - 2x - 3 = (x - 3)(x + 1)$

Numerator 2: $2x^2 - 3x - 9$

Factorize: $2x^2 - 3x - 9 = (2x + 3)(x - 3)$

Denominator 2: $x^2 - x - 2$

Factorize: $x^2 - x - 2 = (x - 2)(x + 1)$

Numerator 3 (Division): $10x^2 + 27x + 18$

Factorize: $10x^2 + 27x + 18 = (5x + 6)(2x + 3)$

Denominator 3 (Division): $x^2 + 2x + 1$

Factorize: $x^2 + 2x + 1 = (x + 1)^2$

Step 2: Rewrite the expression with factorizations.

$\frac{(x + 3)(x - 2)}{(x - 3)(x + 1)} \cdot \frac{(2x + 3)(x - 3)}{(x - 2)(x + 1)} \div \frac{(5x + 6)(2x + 3)}{(x + 1)^2}$

Step 3: Handle the division.

To divide, multiply by the reciprocal: $\frac{(x + 3)(x - 2)}{(x - 3)(x + 1)} \cdot \frac{(2x + 3)(x - 3)}{(x - 2)(x + 1)} \cdot \frac{(x + 1)^2}{(5x + 6)(2x + 3)}$

Step 4: Combine the fractions.

$\frac{(x + 3)(x - 2)(2x + 3)(x - 3)(x + 1)^2}{(x - 3)(x + 1)(x - 2)(x + 1)(5x + 6)(2x + 3)}$

Step 5: Cancel common terms.

• $(x - 3)$ cancels out.
• $(x - 2)$ cancels out.
• One $(x + 1)$ cancels out.
• $(2x + 3)$ cancels out.

The simplified expression is: $\frac{(x + 3)(x + 1)}{5x + 6}$

Final Answer:

$\frac{(x + 3)(x + 1)}{5x + 6}$

Would you like me to explain any specific steps in more detail or provide further clarifications?

Here are some related questions to deepen your understanding:

1. How do you factor quadratic expressions effectively?
2. What are the rules for multiplying and dividing rational expressions?
3. How do you simplify fractions with common terms?
4. What are common mistakes to avoid when dealing with polynomial factorizations?
5. How does the reciprocal work when dividing fractions?

Tip: Always double-check for common terms that can cancel out after factorization.