 
x2 − 49
x2 + 9x + 14
 · 
x2 − 5x − 14
x2 − 7x

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.

Learn how to simplify a complex rational expression involving quadratic polynomials step-by-step. This problem covers factorization of quadratic equations, particularly using the difference of squares formula. Suitable for high school students.

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