To simplify the given expression, we can first factorize each quadratic polynomial in the numerators and denominators. The given expression is:

$\frac{x^2 - 49}{x^2 + 9x + 14} \cdot \frac{x^2 - 5x - 14}{x^2 - 7x}$

Step 1: Factorize each quadratic expression

1. Factorizing $x^2 - 49$: $x^2 - 49 = (x - 7)(x + 7)$ (Difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$)

2. Factorizing $x^2 + 9x + 14$: $x^2 + 9x + 14 = (x + 7)(x + 2)$ (Finding two numbers that multiply to 14 and add to 9: 7 and 2)

3. Factorizing $x^2 - 5x - 14$: $x^2 - 5x - 14 = (x - 7)(x + 2)$ (Finding two numbers that multiply to -14 and add to -5: -7 and 2)

4. Factorizing $x^2 - 7x$: $x^2 - 7x = x(x - 7)$ (Taking $x$ as a common factor)

Step 2: Rewrite the expression using the factorizations

$\frac{(x - 7)(x + 7)}{(x + 7)(x + 2)} \cdot \frac{(x - 7)(x + 2)}{x(x - 7)}$

Step 3: Simplify by canceling common factors

• In the first fraction, $(x + 7)$ cancels out:

  $\frac{(x - 7)}{(x + 2)} \cdot \frac{(x - 7)(x + 2)}{x(x - 7)}$

• In the second fraction, $(x - 7)$ cancels out with one of the $(x - 7)$ in the numerator:

  $\frac{(x - 7)}{(x + 2)} \cdot \frac{(x + 2)}{x}$

• Now, $(x + 2)$ cancels out:

  $\frac{(x - 7)}{x}$

Final Simplified Expression

$\frac{x - 7}{x}$

This is the simplified form of the given expression, valid for $x \neq 0$, $x \neq -7$, and $x \neq 7$ to avoid division by zero.

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Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How do you factorize a quadratic expression in general?
2. What are the conditions for using the difference of squares formula?
3. Why is it important to check for common factors before canceling in a fraction?
4. How can you verify that your simplification is correct?
5. What is the significance of finding the domain in rational expressions?

Tip: Always check for restrictions on the variable values when simplifying rational expressions, to avoid division by zero.