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Let's solve this step by step to find the correct answer.

The given expression is:

\[
\frac{a^2 - a - 6}{a^2 - 8a + 7} \times \frac{a + 7}{a - 3}
\]

### Step 1: Factor the quadratic expressions.

#### First, factor \(a^2 - a - 6\):

To factor \(a^2 - a - 6\), we need two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). Therefore:

\[
a^2 - a - 6 = (a - 3)(a + 2)
\]

#### Next, factor \(a^2 - 8a + 7\):

We need two numbers that multiply to \(7\) and add to \(-8\). These numbers are \(-7\) and \(-1\). So:

\[
a^2 - 8a + 7 = (a - 7)(a - 1)
\]

### Step 2: Substitute the factored forms into the original expression:

Now the expression becomes:

\[
\frac{(a - 3)(a + 2)}{(a - 7)(a - 1)} \times \frac{a + 7}{a - 3}
\]

### Step 3: Simplify the expression.

Notice that \((a - 3)\) appears in both the numerator and denominator, so we can cancel those terms:

\[
\frac{(a + 2)}{(a - 7)(a - 1)} \times \frac{a + 7}{1}
\]

Now, multiply the remaining terms:

\[
\frac{(a + 2)(a + 7)}{(a - 7)(a - 1)}
\]

### Step 4: Compare with the answer choices.

This matches **option A**:

\[
\frac{(a + 2)(a + 7)}{(a - 7)(a - 1)}
\]

### Final Answer:
The correct answer is **A**.

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Would you like more details or have any questions on this problem?

Here are 5 related questions you might find useful:
1. How do you factor more complex quadratics like \(ax^2 + bx + c\)?
2. Can all quadratic equations be factored easily?
3. What are the other methods to simplify rational expressions?
4. How can you check if your factored expression is correct?
5. What happens if an expression can't be factored?

**Tip:** When simplifying algebraic fractions, always check for common factors between numerators and denominators to simplify expressions efficiently.

Simplify the following expression: (a^2 - a - 6)/(a^2 - 8a + 7) * (a + 7)/(a - 3).

Learn how to simplify the rational expression (a^2 - a - 6)/(a^2 - 8a + 7) * (a + 7)/(a - 3) by factoring quadratic expressions and canceling common factors. This problem covers key algebraic concepts, including factoring and simplifying rational expressions. Suitable for students in Grades 8-10.

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