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}\\)?
\\(\circ\\) \\(\frac{(b - 5)(b + 3)}{4}\\)
\\(\circ\\) \\(\frac{b + 3}{8}\\)
\\(\circ\\) \\(\frac{(b + 5)(b - 3)}{4}\\)
\\(\circ\\) \\(\frac{2b + 5}{8}\\)

Explanation:

Step1: Factor numerator and denominators

Factor \(b^2 - 2b - 15\): find two numbers that multiply to -15 and add to -2, which are -5 and 3. So \(b^2 - 2b - 15=(b - 5)(b + 3)\).
Factor \(8b + 20\): take out 4, so \(8b + 20 = 4(2b + 5)\).
Factor \(4b + 10\): take out 2, so \(4b + 10 = 2(2b + 5)\).

The expression becomes \(\frac{(b - 5)(b + 3)}{4(2b + 5)} \div \frac{2}{2(2b + 5)}\).

Step2: Change division to multiplication

Dividing by a fraction is multiplying by its reciprocal:
\(\frac{(b - 5)(b + 3)}{4(2b + 5)} \times \frac{2(2b + 5)}{2}\)

Step3: Cancel common factors

Cancel \(2(2b + 5)\) from numerator and denominator:
\(\frac{(b - 5)(b + 3)}{4} \times \frac{1}{1}=\frac{(b - 5)(b + 3)}{4}\)

Answer:

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