Question

Write the polynomial expression in simplest form: x^2−9/x^2−4⋅4x−8/12x+36
.(1 point)
Responses

(x−3)(x−2)3(x+2)(x+2)
Start Fraction left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis over 3 left parenthesis x plus 2 right parenthesis left parenthesis x plus 2 right parenthesis End Fraction

x−33(x+2)
Start Fraction x minus 3 over 3 left parenthesis x plus 2 right parenthesis End Fraction

13
Start Fraction 1 over 3 End Fraction

2(x−3)6(x+2)

Answer 1

Let's simplify the expression step by step:

The expression is:

\[
\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36}
\]

### Step 1: Factor each part of the expression

1. **Numerator of the first fraction**: \(x^2 - 9\) is a difference of squares.
\[
x^2 - 9 = (x - 3)(x + 3)
\]

2. **Denominator of the first fraction**: \(x^2 - 4\) is also a difference of squares.
\[
x^2 - 4 = (x - 2)(x + 2)
\]

3. **Numerator of the second fraction**: Factor \(4x - 8\):
\[
4x - 8 = 4(x - 2)
\]

4. **Denominator of the second fraction**: Factor \(12x + 36\):
\[
12x + 36 = 12(x + 3)
\]

### Step 2: Rewrite the expression with the factors

Now rewrite the original expression using these factors:
\[
\frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{12(x + 3)}
\]

### Step 3: Combine the fractions

Combine the fractions:
\[
\frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)}
\]

### Step 4: Cancel common factors

Now, we can cancel common factors from the numerator and the denominator:
- \(x - 2\) in the numerator and denominator.
- \(x + 3\) in the numerator and denominator.

This simplifies to:
\[
\frac{4(x - 3)}{12(x + 2)}
\]

### Step 5: Simplify further

Now, simplify the coefficient:
\[
\frac{4}{12} = \frac{1}{3}
\]
So we have:
\[
\frac{x - 3}{3(x + 2)}
\]

### Final Answer

The simplest form of the original expression is:
\[
\frac{x - 3}{3(x + 2)}
\]

Among the options given, the answer matches:
\[
\frac{x - 3}{3(x + 2)}
\]
so the correct choice is:
\[
\frac{x - 3}{3(x + 2)}
\]