To simplify the expression \(\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36}\), we will first factor the polynomials in the numerator and the denominator.
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Factoring \(x^2 - 9\):
- This is a difference of squares: \(x^2 - 9 = (x - 3)(x + 3)\).
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Factoring \(x^2 - 4\):
- This is also a difference of squares: \(x^2 - 4 = (x - 2)(x + 2)\).
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Factoring \(4x - 8\):
- Factor out a 4: \(4x - 8 = 4(x - 2)\).
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Factoring \(12x + 36\):
- Factor out a 12: \(12x + 36 = 12(x + 3)\).
Now we can rewrite the expression using these factorizations:
\[ \frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{12(x + 3)} \]
Next, we can combine this into one fraction:
\[ \frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)} \]
Now we'll cancel out common factors in the numerator and the denominator:
- \(x - 2\) in the numerator and denominator.
- \(x + 3\) in the numerator and denominator.
After canceling these common factors, we are left with:
\[ \frac{4(x - 3)}{12(x + 2)} \]
We can simplify this further by reducing the coefficients:
\[ \frac{4}{12} = \frac{1}{3} \]
Thus, we end up with:
\[ \frac{(x - 3)}{3(x + 2)} \]
Therefore, the final expression in simplest form is:
\[ \frac{x - 3}{3(x + 2)} \]
From the given options, this corresponds to:
\[ \text{Start Fraction left parenthesis x minus 3 right parenthesis over 3 left parenthesis x plus 2 right parenthesis End Fraction} \]