Question

multiply the expressions.
\\(\frac{x^2 - 11x + 30}{x^2 - 2x - 24} \cdot \frac{x^2 - 16}{x^2 - 25}\\)

select the correct choice below and fill in the answer box(es) within your choice.
\\(\bigcirc\\) a. \\(\frac{x^2 - 11x + 30}{x^2 - 2x - 24} \cdot \frac{x^2 - 16}{x^2 - 25} = \square, x \
eq \square\\)
(simplify your answer. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. \\(\frac{x^2 - 11x + 30}{x^2 - 2x - 24} \cdot \frac{x^2 - 16}{x^2 - 25} = \square\\) and no numbers must be excluded.

Explanation:

Step1: Factor all polynomials

1. $x^2 - 11x + 30 = (x-5)(x-6)$
2. $x^2 - 2x - 24 = (x-6)(x+4)$
3. $x^2 - 16 = (x-4)(x+4)$
4. $x^2 - 25 = (x-5)(x+5)$

Step2: Substitute factors into product

$$\frac{(x-5)(x-6)}{(x-6)(x+4)} \cdot \frac{(x-4)(x+4)}{(x-5)(x+5)}$$

Step3: Cancel common factors

Cancel $(x-5)$, $(x-6)$, $(x+4)$ from numerator and denominator:
$$\frac{x-4}{x+5}$$

Step4: Find excluded values

Set original denominators and canceled factors to 0:

1. $x-6=0 \implies x=6$
2. $x+4=0 \implies x=-4$
3. $x-5=0 \implies x=5$
4. $x+5=0 \implies x=-5$

Answer:

A. $\frac{x^2 - 11x + 30}{x^2 - 2x - 24} \cdot \frac{x^2 - 16}{x^2 - 25} = \frac{x-4}{x+5}, x
eq -5, -4, 5, 6$