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

1 answer

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.