Problem 30 - College Algebra

Question

Multiply or divide as indicated. $$\frac{x^{2}-4}{x^{2}+3 x-10} \div \frac{x^{2}+5 x+6}{x^{2}+8 x+15}$$

Step-by-Step Solution

Answer

The result of this operation is \[\frac{x+2}{x+5}\].

Step 1: Simplify

First, rewrite the division as multiplication by flipping (writing the reciprocal) the second term. Now, the operation becomes: \[\frac{x^{2}-4}{x^{2}+3 x-10} \times \frac{x^{2}+8 x+15}{x^{2}+5 x+6}\].

Step 2: Factor

For simplification, factor all the expressions. The expression \(x^{2}-4\) can be factored as \((x-2)(x+2)\), \(x^{2}+3x-10\) factors to \((x-2)(x+5)\) and \(x^{2}+8x+15\) factors to \((x+3)(x+5)\). Finally, \(x^{2}+5x+6\) factors to \((x+2)(x+3)\). Insert the factors into the expression: \[\frac{(x-2)(x+2)}{(x-2)(x+5)} \times \frac{(x+3)(x+5)}{(x+2)(x+3)}\].

Step 3: Cancel and Multiply

Next, cancel out the common factors in the numerator and the denominator. After the cancellation, multiply the remaining factors in the numerator to get the numerator of the answer, and do the same for the denominator. The remaining factors are \((x+2)\) in the numerator and \((x+5)\) in the denominator. Thus, the final answer is \[\frac{x+2}{x+5}\].