To simplify the given expression
\[ \left( \frac{2x^2 - 5x - 3}{x^2 + x - 20} \right) \left( \frac{x^2 + 8x + 15}{2x^2 + 9x + 4} \right), \]
we need to factor all the polynomials involved.
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Factoring \(2x^2 - 5x - 3\):
To factor \(2x^2 - 5x - 3\), we look for two numbers that multiply to \(2 \times -3 = -6\) and add to \(-5\). The numbers are \(-6\) and \(1\). \[ 2x^2 - 6x + x - 3 = 2x(x - 3) + 1(x - 3) = (2x + 1)(x - 3). \]
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Factoring \(x^2 + x - 20\):
For \(x^2 + x - 20\), we want two numbers that multiply to \(-20\) and add up to \(1\). The numbers are \(5\) and \(-4\). \[ x^2 + 5x - 4x - 20 = (x + 5)(x - 4). \]
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Factoring \(x^2 + 8x + 15\):
Here we need two numbers that multiply to \(15\) and add to \(8\). The numbers are \(3\) and \(5\). \[ x^2 + 8x + 15 = (x + 3)(x + 5). \]
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Factoring \(2x^2 + 9x + 4\):
Lastly, for \(2x^2 + 9x + 4\), we find two numbers that multiply to \(2 \times 4 = 8\) and add to \(9\). The numbers are \(8\) and \(1\). \[ 2x^2 + 8x + x + 4 = 2x(x + 4) + 1(x + 4) = (2x + 1)(x + 4). \]
Now, substituting these factors back into the expression, we have:
\[ \frac{(2x + 1)(x - 3)}{(x + 5)(x - 4)} \cdot \frac{(x + 3)(x + 5)}{(2x + 1)(x + 4)}. \]
Next, we can cancel out the common factors:
- \( (2x + 1) \)
- \( (x + 5) \)
Thus, what remains is:
\[ \frac{(x - 3)(x + 3)}{(x - 4)(x + 4)}. \]
Now, we notice that \( (x - 3)(x + 3) = x^2 - 9 \) and \( (x - 4)(x + 4) = x^2 - 16 \).
So the simplified expression is:
\[ \frac{x^2 - 9}{x^2 - 16}. \]
Thus, the answer is
\[ \boxed{\frac{x^2 - 9}{x^2 - 16}}. \]