Question

Multiply the rational expressions and simplify.

fraction with numerator x squared minus 9 and denominator x squared minus x minus 6 times fraction with numerator x squared minus 4 and denominator x plus 3

x + 3
x + 2
x – 2
x – 3

Answer 1

To multiply the given rational expressions, we will first write them out clearly and then simplify.

The two rational expressions are:

\[
\frac{x^2 - 9}{x^2 - x - 6} \times \frac{x^2 - 4}{x + 3}
\]

### Step 1: Factor the numerators and denominators

**Numerator 1: \(x^2 - 9\)**
This is a difference of squares and can be factored as:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

**Denominator 1: \(x^2 - x - 6\)**
To factor this, we look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\):
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

**Numerator 2: \(x^2 - 4\)**
This is another difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

Now we can rewrite our expression with the factored forms:
\[
\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)} \times \frac{(x - 2)(x + 2)}{(x + 3)}
\]

### Step 2: Multiply the expressions

Now we multiply the numerators and the denominators:
\[
\frac{(x - 3)(x + 3)(x - 2)(x + 2)}{(x - 3)(x + 2)(x + 3)}
\]

### Step 3: Cancel common factors

Now we can cancel out the common factors in the numerator and denominator:

- \(x + 3\) in the numerator and denominator.
- \(x - 3\) in the numerator and denominator.

This leaves us with:
\[
\frac{(x - 2)(x + 2)}{(x + 2)}
\]

We can cancel \(x + 2\) as well (provided \(x \neq -2\)):
\[
\frac{x - 2}{1} = x - 2
\]

### Final Result

Thus, the simplified form of the expression is:
\[
x - 2
\]

This matches the answer among your options:
1. \(x + 3\)
2. \(x + 2\)
3. \(x - 2\) (This is the correct answer)
4. \(x - 3\)

So, the correct answer is:
\[
\boxed{x - 2}
\]