How do you simplify #(2x^2+5x+2)/(4x^2-1) xx (2x^2+x-1)/(x^2+x-2)#?

Question

Avery Alexander · Accepted Answer

#(x+1)/(x-1)# Factor. #color(red)(2x^2+5x+2)/color(blue)(4x^2-1) * color(purple)(2x^2+x-1)/color(green)(x^2+x-2)# #color(red)((2x+1)(x+2))/color(blue)((2x-1)(2x+1)) * color(purple)((2x-1)(x+1))/color(green)((x+2)(x-1))# Cancel. #(color(red)((2x+1))color(orange)((x+2)))/(color(blue)((2x-1)color(red)((2x+1)))) * color(blue)((2x-1)color(black)((x+1)))/(color(orange)((x+2)color(black)((x-1))# #(cancelcolor(red)((2x+1))cancelcolor(orange)((x+2)))/(cancel(color(blue)((2x-1)))cancelcolor(red)((2x+1))) * (cancelcolor(blue)((2x-1))color(black)((x+1)))/(cancel(color(orange)((x+2)))color(black)((x-1))# Simplify. #(x+1)/(x-1)#

Avery Alexander · Answer

undefined To simplify the expression (2x^2+5x+2)/(4x^2-1) × (2x^2+x-1)/(x^2+x-2), we can follow these steps:

1. Factorize the denominators:
   4x^2-1 can be factored as (2x+1)(2x-1)
   x^2+x-2 can be factored as (x+2)(x-1)

2. Rewrite the expression with the factored denominators:
   (2x^2+5x+2)/[(2x+1)(2x-1)] × (2x^2+x-1)/[(x+2)(x-1)]

3. Cancel out any common factors between the numerators and denominators:
   (2x+1) and (2x-1) cancel out in the first fraction
   (x-1) cancels out in the second fraction

4. Simplify the expression:
   (2x+1) × (2x^2+x-1)/(x+2)

Therefore, the simplified expression is (2x+1) × (2x^2+x-1)/(x+2).