Question

Question
Simplify the expression: left parenthesis Start Fraction 2 x squared minus 5 x minus 3 over x squared plus x minus 20 End Fraction right parenthesis left parenthesis Start Fraction x squared plus 8 x plus 15 over 2 x squared plus 9 x plus 4 End Fraction right parenthesis
(1 point)
Responses

Start Fraction x squared minus 9 over x squared minus 16 End Fraction
Image with alt text: Start Fraction x squared minus 9 over x squared minus 16 End Fraction

Start Fraction 9 over 16 End Fraction
Image with alt text: Start Fraction 9 over 16 End Fraction

Start Fraction x squared minus 6 x plus 9 over x squared minus 8 x plus 16 End Fraction
Image with alt text: Start Fraction x squared minus 6 x plus 9 over x squared minus 8 x plus 16 End Fraction

Start Fraction x squared plus 6 x plus 9 over x squared plus 8 x plus 16 End Fraction

Answer 1

To simplify the expression

\[
\left( \frac{2x^2 - 5x - 3}{x^2 + x - 20} \right) \left( \frac{x^2 + 8x + 15}{2x^2 + 9x + 4} \right),
\]

we will factor each polynomial in the numerators and denominators where possible.

**Step 1: Factor the polynomials**

1. For \(2x^2 - 5x - 3\):
\[
2x^2 - 5x - 3 = (2x + 1)(x - 3)
\]

2. For \(x^2 + x - 20\):
\[
x^2 + x - 20 = (x + 5)(x - 4)
\]

3. For \(x^2 + 8x + 15\):
\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\]

4. For \(2x^2 + 9x + 4\):
\[
2x^2 + 9x + 4 = (2x + 1)(x + 4)
\]

**Step 2: Rewrite the expression with the factored forms**

Substituting in the factored forms, we have:

\[
\left( \frac{(2x + 1)(x - 3)}{(x + 5)(x - 4)} \right) \left( \frac{(x + 3)(x + 5)}{(2x + 1)(x + 4)} \right)
\]

**Step 3: Multiply the fractions**

Now, combine the numerators and the denominators:

\[
\frac{(2x + 1)(x - 3)(x + 3)(x + 5)}{(x + 5)(x - 4)(2x + 1)(x + 4)}
\]

**Step 4: Cancel common factors**

We can cancel the common factors:

- \( (2x + 1) \) cancels with \( (2x + 1) \)
- \( (x + 5) \) cancels with \( (x + 5) \)

After cancellation, we have:

\[
\frac{(x - 3)(x + 3)}{(x - 4)(x + 4)}
\]

**Step 5: Recognize the resultant expression**

The numerator can be simplified further:

\[
(x - 3)(x + 3) = x^2 - 9
\]

The denominator can be addressed as:

\[
(x - 4)(x + 4) = x^2 - 16
\]

Thus, our simplified expression is:

\[
\frac{x^2 - 9}{x^2 - 16}
\]

**Final Answer:**

So the resulting simplified expression is

\[
\frac{x^2 - 9}{x^2 - 16}.
\]

This corresponds to the first response option.