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The problem asks to simplify the rational expression:

\[
\frac{x^2 - 9x + 14}{x^2 + 5x - 14}
\]

Let's first factor both the numerator and the denominator:

### Numerator: \(x^2 - 9x + 14\)
We need to factor this quadratic expression. Look for two numbers that multiply to 14 and add to -9. These numbers are -7 and -2. Thus, we can factor it as:

\[
x^2 - 9x + 14 = (x - 7)(x - 2)
\]

### Denominator: \(x^2 + 5x - 14\)
Now factor this quadratic expression. Look for two numbers that multiply to -14 and add to 5. These numbers are 7 and -2. Thus, we can factor it as:

\[
x^2 + 5x - 14 = (x + 7)(x - 2)
\]

### Simplifying the expression:
Now, substitute the factored forms back into the original expression:

\[
\frac{(x - 7)(x - 2)}{(x + 7)(x - 2)}
\]

Since \(x - 2\) appears in both the numerator and denominator, they cancel out:

\[
\frac{x - 7}{x + 7}
\]

So, the simplified form is:

\[
\frac{x - 7}{x + 7}
\]

This matches the second option in the list. Therefore, the correct answer is:

\[
\boxed{\frac{x - 7}{x + 7}}
\]

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. How do you factor quadratic expressions in general?
2. What happens if the quadratic in the numerator and the denominator have no common factors?
3. How can you simplify more complex rational expressions involving higher powers?
4. Why is it important to check for restrictions on the variable after simplification?
5. What is the difference between simplifying and solving a rational expression?

**Tip:** When simplifying rational expressions, always look for common factors in the numerator and denominator before cancelling them out.

Simplify the rational expression: (x^2 - 9x + 14) / (x^2 + 5x - 14)

Learn how to simplify the rational expression (x^2 - 9x + 14) / (x^2 + 5x - 14) using quadratic factorization. This step-by-step solution involves factoring both the numerator and denominator and canceling out common terms. Ideal for high school students in Grades 9-10.

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