To simplify the expression \( \frac{4x + 8}{3x + 18} \div \frac{x + 2}{x^2 - 36} \), we can follow these steps:
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Rewrite the division as multiplication by the reciprocal: \[ \frac{4x + 8}{3x + 18} \times \frac{x^2 - 36}{x + 2} \]
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Factor all expressions:
- \( 4x + 8 = 4(x + 2) \)
- \( 3x + 18 = 3(x + 6) \)
- \( x^2 - 36 = (x - 6)(x + 6) \) (difference of squares)
- \( x + 2 \) is already factored.
Now, substituting in the factored forms: \[ \frac{4(x + 2)}{3(x + 6)} \times \frac{(x - 6)(x + 6)}{x + 2} \]
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Cancel out the \( x + 2 \) from the numerator and the denominator: \[ \frac{4}{3(x + 6)} \times (x - 6)(x + 6) \]
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This simplifies to: \[ \frac{4(x - 6)(x + 6)}{3(x + 6)} \]
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We can cancel \( x + 6 \) in the numerator and denominator, assuming \( x + 6 \neq 0 \): \[ \frac{4(x - 6)}{3} \]
So, the final simplified expression is: \[ \frac{4(x - 6)}{3} \]
Now, let's compare with the given options:
- A. \( 4(x−2) \)
- B. \( 4x−24 \)
- C. \( 4x−8 \)
- D. \( \frac{3}{4}(x−6) \)
The expression we found, \( \frac{4(x - 6)}{3} \), matches fundamentally with Option D, which can be rewritten as \( 4 \frac{(x - 6)}{3} \), indicating they express the same value structurally.
Thus, the answer is: \[ \boxed{D} \]