Question

Question was provided via image upload.

Explanation:

Step1: Factor each polynomial

First, factor the numerators and denominators:

• For \(4x^2 - 16\), factor out 4: \(4(x^2 - 4)=4(x - 2)(x + 2)\) (using difference of squares \(a^2 - b^2=(a - b)(a + b)\))
• For \(x^2 + 5x + 6\), factor: \((x + 2)(x + 3)\) (finding two numbers that multiply to 6 and add to 5)
• For \(x^2 - 7x - 30\), factor: \((x - 10)(x + 3)\) (finding two numbers that multiply to -30 and add to -7)
• For \(16x^2 - 192x + 320\), factor out 16: \(16(x^2 - 12x + 20)=16(x - 10)(x - 2)\) (finding two numbers that multiply to 20 and add to -12)

Step2: Substitute factored forms into the expression

Substitute these factored forms into the original expression:

$$ \frac{4(x - 2)(x + 2)}{(x + 2)(x + 3)} \times \frac{(x - 10)(x + 3)}{16(x - 10)(x - 2)} $$

Step3: Cancel out common factors

Cancel out the common factors:

• \(4\) and \(16\) have a common factor of \(4\), so \(\frac{4}{16}=\frac{1}{4}\)
• \((x - 2)\) cancels out from numerator and denominator
• \((x + 2)\) cancels out
• \((x + 3)\) cancels out
• \((x - 10)\) cancels out

After canceling, we are left with \(\frac{1}{4}\)

Answer:

D. \(\frac{1}{4}\)