Question

divide.
\\(\frac{x^{2}+x}{x^{2}-16} \div \frac{x^{2}-1}{x^{2}+10x + 24}\\)

\\(\frac{x^{2}+x}{x^{2}-16} \div \frac{x^{2}-1}{x^{2}+10x + 24}=\square\\), where \\(x\
eq\square\\)
(simplify your answer. type your answer in factored form. use a comma to s

Explanation:

Step1: Rewrite division as multiplication

$\frac{x^2+x}{x^2-16} \times \frac{x^2+10x+24}{x^2-1}$

Step2: Factor all polynomials

$\frac{x(x+1)}{(x-4)(x+4)} \times \frac{(x+4)(x+6)}{(x-1)(x+1)}$

Step3: Cancel common factors

$\frac{x\cancel{(x+1)}}{(x-4)\cancel{(x+4)}} \times \frac{\cancel{(x+4)}(x+6)}{(x-1)\cancel{(x+1)}}$

Step4: Multiply remaining terms

$\frac{x(x+6)}{(x-4)(x-1)}$

Step5: Find excluded values

Identify $x$ that make original denominators/ divisor zero: $x
eq -6, -4, -1, 1, 4$

Answer:

$\frac{x(x+6)}{(x-4)(x-1)}$, where $x
eq -6, -4, -1, 1, 4$