Fraction 1: the quantity x squared minus 16 over the quantity 2 times x squared minus 9 times x plus 4; Fraction 2: the quantity 2 times x squared plus 14 times x plus 24 over the quantity 4 times x plus 4; Find Fraction 1 divided by Fraction 2.

Answer 1

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, we need to find the reciprocal of fraction 2:

Reciprocal of Fraction 2:

- Flip the fraction (upside down): the quantity 4 times x plus 4 over the quantity 2 times x squared plus 14 times x plus 24.
- Simplify by canceling out common factors: the quantity x plus 1 over the quantity x squared plus 7 times x plus 12.

Now, we can multiply fraction 1 by the reciprocal of fraction 2:

Fraction 1 divided by Fraction 2:

- Multiply the numerators: (x squared minus 16) times (x squared plus 7 times x plus 12).
- Multiply the denominators: (2 times x squared minus 9 times x plus 4) times (x plus 1).
- Simplify the numerator by factoring: (x - 4) times (x + 4) times (x + 3) times (x + 1).
- Cancel out common factors in the numerator and denominator: (x - 4) times (x + 3) divided by (2 times x squared - 9 times x + 4).

Therefore, Fraction 1 divided by Fraction 2 is:

(x squared - 16) times (x squared + 7 times x + 12)
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(2 times x squared - 9 times x + 4) times (x + 1)

= (x - 4) times (x + 4) times (x + 3) times (x + 1)
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(2x - 1) times (x + 4)

Answer: (x - 4) times (x + 3) times (x + 1) divided by (2x - 1) times (x + 4).