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The question in your image asks you to perform operations and simplify the following expression:

\[
\frac{x^2 + 2x - 15}{x^2 - x - 6} \cdot \frac{3x^2 - 10x + 8}{x^2 + 7x + 10} \div \frac{6x^2 + 7x + 2}{x^2 + 4x + 4}
\]

Let's break this down step by step:

### 1. Factor each expression:
- **First fraction:**
  \[
  \frac{x^2 + 2x - 15}{x^2 - x - 6}
  \]
  - Numerator: \(x^2 + 2x - 15 = (x - 3)(x + 5)\)
  - Denominator: \(x^2 - x - 6 = (x - 3)(x + 2)\)

  So, this fraction becomes:
  \[
  \frac{(x - 3)(x + 5)}{(x - 3)(x + 2)}
  \]

- **Second fraction:**
  \[
  \frac{3x^2 - 10x + 8}{x^2 + 7x + 10}
  \]
  - Numerator: \(3x^2 - 10x + 8 = (3x - 4)(x - 2)\)
  - Denominator: \(x^2 + 7x + 10 = (x + 2)(x + 5)\)

  So, this fraction becomes:
  \[
  \frac{(3x - 4)(x - 2)}{(x + 2)(x + 5)}
  \]

- **Third fraction:**
  \[
  \frac{6x^2 + 7x + 2}{x^2 + 4x + 4}
  \]
  - Numerator: \(6x^2 + 7x + 2 = (3x + 2)(2x + 1)\)
  - Denominator: \(x^2 + 4x + 4 = (x + 2)^2\)

  So, this fraction becomes:
  \[
  \frac{(3x + 2)(2x + 1)}{(x + 2)^2}
  \]

### 2. Rewrite the expression:
The division of fractions can be turned into multiplication by taking the reciprocal of the third fraction:
\[
\frac{(x - 3)(x + 5)}{(x - 3)(x + 2)} \cdot \frac{(3x - 4)(x - 2)}{(x + 2)(x + 5)} \cdot \frac{(x + 2)^2}{(3x + 2)(2x + 1)}
\]

### 3. Simplify:
Now, cancel out common terms across the numerators and denominators:
- \(x - 3\) cancels out from the first fraction.
- \(x + 5\) cancels out from both the second and third fractions.
- \(x + 2\) appears in multiple terms, but we have \((x + 2)^2\) in the numerator of the third fraction, and two \(x + 2\)'s in the denominator.

After canceling, we're left with:
\[
\frac{(3x - 4)(x - 2)}{(3x + 2)(2x + 1)}
\]

This is the simplified form of the given expression.

Would you like more details or further explanation?

Here are five related questions for further practice:
1. How do you factor quadratic expressions with different leading coefficients?
2. How do you perform multiplication and division of rational expressions?
3. Can you simplify rational expressions involving higher powers, such as cubes?
4. What are the steps to handle complex fractions in algebra?
5. How do you recognize common factors to simplify before multiplying fractions?

**Tip:** Always factor each expression fully before multiplying or dividing to easily identify and cancel out common factors.

Perform the given operations and simplify: (x^2 + 2x - 15) / (x^2 - x - 6) * (3x^2 - 10x + 8) / (x^2 + 7x + 10) ÷ (6x^2 + 7x + 2) / (x^2 + 4x + 4)

Learn how to simplify a complex rational expression involving multiplication and division of quadratic equations. This detailed explanation covers factoring quadratics, multiplying and dividing rational expressions, and simplifying common factors. Perfect for high school algebra students.

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