Question

divide: \\(\frac{x^2 + 7x + 12}{x^2 + 2x - 24} \div \frac{x^2 - 16}{x^2 + 9x + 18}\\)
a. \\(\frac{(x + 3)(x + 6)}{(x + 4)(x - 4)}\\)
b. \\(1\\)
c. \\(\frac{(x + 4)^2}{(x + 6)^2}\\)
d. \\(\frac{(x + 3)^2}{(x - 4)^2}\\)

Explanation:

Step1: Factor all quadratics

$x^2+7x+12=(x+3)(x+4)$; $x^2+2x-24=(x+6)(x-4)$; $x^2-16=(x+4)(x-4)$; $x^2+9x+18=(x+3)(x+6)$

Step2: Rewrite division as multiplication

$\frac{(x+3)(x+4)}{(x+6)(x-4)} \times \frac{(x+3)(x+6)}{(x+4)(x-4)}$ → Cancel common factors: $(x+4),(x+6)$

Step3: Simplify remaining terms

$\frac{(x+3)(x+6)}{(x+4)(x-4)}$

Answer:

A. $\frac{(x+3)(x+6)}{(x+4)(x-4)}$