Chapter A.5, Problem 33AYU

To determine

To find: Simplify the given in factored form.

Answer to Problem 33AYU

$\frac{-2x\left(x^2-2\right)}{\left(x+2\right)\left(x^2-x-3\right)}$

Explanation of Solution

Given:

$\frac{\frac{x-2}{x+2}+\frac{x-1}{x+1}}{\frac{x}{x+1}-\frac{2x-3}{x}}$

Calculation:

$\frac{\frac{x-2}{x+2}+\frac{x-1}{x+1}}{\frac{x}{x+1}-\frac{2x-3}{x}}=\frac{\frac{x-2}{x+2}+\frac{x-1}{x+1}}{\frac{x}{x+1}-\frac{2x-3}{x}}=\frac{\frac{x-2\left(x+1\right)+x-1\left(x+2\right)}{\left(x+2\right)\left(x+1\right)}}{\frac{x\left(x\right)-\left(2x-3\right)\left(x+1\right)}{x\left(x+1\right)}}$ $=\frac{\left(x-2\right)\left(x+1\right)+\left(x-1\right)\left(x+2\right)}{\left(x+2\right)\left(x+1\right)}.\frac{x\left(x+1\right)}{x\left(x\right)-\left(2x-3\right)\left(x+1\right)}=\frac{-2x\left(x^2-2\right)}{\left(x+2\right)\left(x^2-x-3\right)}$