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Let's simplify the given expression step by step:

The expression is:
\[
\frac{x^2 - 25}{x^2 + 7x + 10}
\]

### Step 1: Factor the numerator and denominator.

1. **Numerator:** \( x^2 - 25 \) is a difference of squares, which can be factored as:
   \[
   x^2 - 25 = (x + 5)(x - 5)
   \]

2. **Denominator:** \( x^2 + 7x + 10 \) can be factored by finding two numbers that multiply to 10 and add up to 7, which are 5 and 2:
   \[
   x^2 + 7x + 10 = (x + 5)(x + 2)
   \]

### Step 2: Rewrite the original expression with these factored forms:
\[
\frac{(x + 5)(x - 5)}{(x + 5)(x + 2)}
\]

### Step 3: Cancel the common factor \((x + 5)\) from the numerator and denominator, assuming \(x \neq -5\):
\[
\frac{x - 5}{x + 2}
\]

Thus, the simplified form is:
\[
\frac{x - 5}{x + 2}
\]

The correct answer is **A. \(\frac{x - 5}{x + 2}\)**.

Would you like a detailed explanation of any step? Here are some follow-up questions that can help deepen your understanding:

1. Why is it important to factor expressions when simplifying fractions?
2. What are some common factoring techniques used for quadratic expressions?
3. How do you recognize a difference of squares and factor it quickly?
4. Why must we exclude certain values of \(x\) when simplifying fractions?
5. How can factoring help in solving equations?

**Tip:** When simplifying rational expressions, always look for common factors to cancel, but remember to state any restrictions on the variable.

Simplify \(\frac{x^2 - 25}{x^2 + 7x + 10}\) to lowest terms.

Learn how to simplify the rational expression \(\frac{x^2 - 25}{x^2 + 7x + 10}\) by factoring both the numerator and denominator, applying the difference of squares and factoring quadratic techniques. This problem is suitable for students in Grades 9-10.

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