Problem 30 - College Algebra Essentials

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 simplified form of the given expression is \( \frac{(x-2)(x+5)}{(x-5)(x+2)} \).

Step 1: Rewriting the Divisor

Change the equation to multiplication by taking the reciprocal of the divisor. \[\frac{x^{2}-4}{x^{2}+3 x-10} \times \frac{x^{2}+8 x+15}{x^{2}+5 x+6}\]

Step 2: Factorize

Factorize all the expressions. Doing this helps to simplify the expression. We get:\[\frac{(x-2)(x+2)}{(x-5)(x+2)} \times \frac{(x+3)(x+5)}{(x+2)(x+3)}\]

Step 3: Simplify

Now cancel out common factors:\[\frac{(x-2)}{(x-5)} \times \frac{(x+5)}{(x+2)}\]

Step 4: Multiply

Finally, multiply the remaining fractions:\[\frac{(x-2)(x+5)}{(x-5)(x+2)}\]This cannot be further simplified.