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
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For \(2x^2 - 5x - 3\): \[ 2x^2 - 5x - 3 = (2x + 1)(x - 3) \]
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For \(x^2 + x - 20\): \[ x^2 + x - 20 = (x + 5)(x - 4) \]
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For \(x^2 + 8x + 15\): \[ x^2 + 8x + 15 = (x + 3)(x + 5) \]
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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.