Question

question 5 of 10
select the correct answer.
if no denominator equals zero, which expression is equivalent to \\(\frac{b^2 - 2b - 15}{8b + 20} \div \frac{2}{4b + 10}\\)?
\\(\frac{(b - 5)(b + 3)}{4}\\)
\\(\frac{b + 3}{8}\\)
\\(\frac{(b + 5)(b - 3)}{4}\\)
\\(\frac{2b + 5}{8}\\)

Explanation:

Step1: Rewrite division as multiplication

To divide by a fraction, multiply by its reciprocal. So, $\frac{b^2 - 2b - 15}{8b + 20} \div \frac{2}{4b + 10}$ becomes $\frac{b^2 - 2b - 15}{8b + 20} \times \frac{4b + 10}{2}$.

Step2: Factor numerators and denominators

• Factor the quadratic in the numerator: $b^2 - 2b - 15 = (b - 5)(b + 3)$ (since $-5 \times 3 = -15$ and $-5 + 3 = -2$).
• Factor the denominator $8b + 20$: $8b + 20 = 4(2b + 5)$.
• Factor the numerator $4b + 10$: $4b + 10 = 2(2b + 5)$.

Now the expression is $\frac{(b - 5)(b + 3)}{4(2b + 5)} \times \frac{2(2b + 5)}{2}$.

Step3: Cancel common factors

• Cancel out the common factor of $2$ in the numerator and denominator of the second fraction: $\frac{(b - 5)(b + 3)}{4(2b + 5)} \times \frac{(2b + 5)}{1}$.
• Cancel out the common factor of $(2b + 5)$: $\frac{(b - 5)(b + 3)}{4} \times \frac{1}{1}$.

Step4: Simplify the expression

After canceling, we are left with $\frac{(b - 5)(b + 3)}{4}$.

Answer:

$\frac{(b - 5)(b + 3)}{4}$ (corresponding to the first option: $\boldsymbol{\frac{(b - 5)(b + 3)}{4}}$)