Final answer:
The quotient of the given rational expressions is found by factoring and then multiplying by the reciprocal of the second fraction. Common terms are cancelled out, leaving the reduced form 4(x - 2)/(x + 2). Thus, the correct answer is C. 4(x - 2)/(x + 2).
Step-by-step explanation:
To find the quotient of the rational expressions provided, we first need to divide the two expressions given. The question is asking us to divide x2 - 4 by x + 3 and then to divide that result by the division of x2 + 4x + 4 by 4x + 12. To do this, we will multiply the first expression by the reciprocal of the second expression.
First, we factor the numerators and denominators where possible:
- x2 - 4 can be factored to (x + 2)(x - 2) (difference of squares).
- x2 + 4x + 4 can be factored to (x + 2)(x + 2) or (x + 2)2 (a perfect square).
- 4x + 12 can be factored to 4(x + 3).
Now our expression looks like:
(x + 2)(x - 2) / (x + 3) / (x + 2)2 / 4(x + 3)
Next, we multiply by the reciprocal:
((x + 2)(x - 2) / (x + 3)) * (4(x + 3) / (x + 2)2)
We then cancel out the common terms:
- (x + 3) in the denominator and numerator cancel out.
- One (x + 2) from (x + 2)2 and the (x + 2) in the numerator cancel out.
The reduced form of the expression is:
4(x - 2) / (x + 2)
Therefore, the correct answer is C. 4(x - 2)/(x + 2).