To simplify the expression \( \frac{x^2 - 6x - 16}{x^2 - 64} \div \frac{2x + 4}{x^2 + x - 56} \), we can start by rewriting the division as multiplication by the reciprocal.
This gives us:
\[ \frac{x^2 - 6x - 16}{x^2 - 64} \times \frac{x^2 + x - 56}{2x + 4} \]
Next, let's factor each component of the expression.
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Factor \( x^2 - 6x - 16 \): \[ x^2 - 6x - 16 = (x - 8)(x + 2) \]
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Factor \( x^2 - 64 \) (difference of squares): \[ x^2 - 64 = (x - 8)(x + 8) \]
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Factor \( 2x + 4 \): \[ 2x + 4 = 2(x + 2) \]
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Factor \( x^2 + x - 56 \): \[ x^2 + x - 56 = (x - 7)(x + 8) \]
Now substituting these factored forms back into the expression, we get:
\[ \frac{(x - 8)(x + 2)}{(x - 8)(x + 8)} \times \frac{(x - 7)(x + 8)}{2(x + 2)} \]
Now, we can simplify step-by-step:
- Cancel \( (x - 8) \) from the numerator and the denominator.
- Cancel \( (x + 2) \) from the numerator and the denominator.
This leaves us with:
\[ \frac{(x - 7)(x + 8)}{2} \]
Now, this can be rewritten as:
\[ \frac{(x - 7)(x + 8)}{2} \]
Thus, the simplest form of the original expression is:
\[ \frac{(x - 7)(x + 8)}{2} \]
Since this matches the second response option given (the fraction with numerator \( x - 7 \) and denominator \( 2 \)), the correct answer is:
the fraction with numerator \( x - 7 \) and denominator \( 2 \).