Question

multiple choice 1 point which expression is equivalent to \\(dfrac{x^2 - 3x - 10}{x^2 - 2x - 15} \div dfrac{x + 2}{x^2 + 7x + 12}\\)? \\(x + 4\\) \\(dfrac{x + 4}{x + 2}\\) \\(1\\) \\(dfrac{x + 10}{x + 9}\\)

Explanation:

Step1: Factor numerators and denominators

Factor \(x^2 - 3x - 10\): \(x^2 - 3x - 10=(x - 5)(x + 2)\)
Factor \(x^2 - 2x - 15\): \(x^2 - 2x - 15=(x - 5)(x + 3)\)
Factor \(x^2 + 7x + 12\): \(x^2 + 7x + 12=(x + 3)(x + 4)\)
The expression becomes \(\frac{(x - 5)(x + 2)}{(x - 5)(x + 3)}\div\frac{x + 2}{(x + 3)(x + 4)}\)

Step2: Change division to multiplication

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

Step3: Cancel common factors

Cancel \((x - 5)\), \((x + 2)\), and \((x + 3)\) from numerator and denominator:
After canceling, we get \(x + 4\)

Answer:

\(x + 4\) (the option with this expression)