To simplify the expression \( \frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{2x + 36} \), we first need to factor the polynomials involved in the expression.
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Factoring:
- For \( x^2 - 9 \), this is a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3) \]
- For \( x^2 - 4 \), this is also a difference of squares: \[ x^2 - 4 = (x - 2)(x + 2) \]
- For \( 4x - 8 \), we can factor out a 4: \[ 4x - 8 = 4(x - 2) \]
- For \( 2x + 36 \), we can factor out a 2: \[ 2x + 36 = 2(x + 18) \]
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Substituting the factors back into the expression: We can now rewrite the expression: \[ \frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{2(x + 18)} \]
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Combine the fractions: This leads to: \[ \frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 2(x + 18)} \]
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Canceling common factors: We can cancel \( (x - 2) \) from the numerator and the denominator: \[ \frac{(x - 3)(x + 3) \cdot 4}{(x + 2) \cdot 2(x + 18)} \] Simplifying gives: \[ \frac{2(x - 3)(x + 3)}{(x + 2)(x + 18)} \]
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Result in simplest form: The simplified polynomial expression is: \[ \frac{2(x - 3)(x + 3)}{(x + 2)(x + 18)} \]
Now, looking for the closest match among your possible responses, it appears that the first response is the most similar but not perfectly matching. The expression we derived is not simplistically represented as any of those options (none appears quite exact to our simplified term).
The closest analogue of our result can be taken from:
- \( \frac{2(x - 3)}{6(x + 2)} \)
This factor allows for a general examination of \(x\) and its treatment equally divided as stated previously but also denotes some divergence of denominational facilitation highlighting this.
So if we are to frame this, we can denote options or further response based purely on given form.
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