How do you simplify #(2x + 10) /( x - 1) times (x^2 - 1) /( x + 5) - 4 /(x + 1)#?

Question

How do you simplify #(2x + 10) /( x - 1)   times   (x^2 - 1) /( x + 5)  - 4 /(x + 1)#?

Abigail Allen · Accepted Answer

#=(2(x+1)^2 -4)/(x+1)#

#=(2(x^2+2x-1))/((x+1))# Factorise wherever possible. #(2x + 10) /( x - 1) times (x^2 - 1) /( x + 5) - 4 /(x + 1)" "larr# there are 2 terms #=color(blue)((2(x+5))/((x-1)) xx ((x+1)(x-1))/((x+5))) -color(red)(4/((x+1)))# You may cancel in a term, as long as it is through a #xx# sign: #=color(blue)((2cancel((x+5)))/cancel((x-1)) xx ((x+1)cancel((x-1)))/cancel((x+5))) -color(red)(4/((x+1)))# #=color(blue)((2(x+1))/1) - color(red)(4/((x+1)))" "larr# find LCD and subtract #=(2(x+1)(x+1)-4)/((x+1)) # #=(2(x+1)^2 -4)/(x+1)# Expanding further gives: #(2(x^2+2x+1)-4)/((x+1))# #=(2x^2+4x+2-4)/((x+1))# #=(2x^2+4x-2)/((x+1))# #=(2(x^2+2x-1))/((x+1))#

Abigail Allen · Answer

undefined To simplify the expression (2x + 10) / (x - 1) * (x^2 - 1) / (x + 5) - 4 / (x + 1), we can follow these steps:

1. Factorize where possible:
   (2x + 10) / (x - 1) = 2(x + 5) / (x - 1)
   (x^2 - 1) = (x + 1)(x - 1)

2. Rewrite the expression:
   2(x + 5) / (x - 1) * (x + 1)(x - 1) / (x + 5) - 4 / (x + 1)

3. Cancel out common factors:
   (2 * 1) / 1 * (x + 5) / 1 - 4 / (x + 1)

4. Simplify further:
   2(x + 5) - 4 / (x + 1)

5. Distribute and combine like terms:
   2x + 10 - 4 / (x + 1)

Therefore, the simplified expression is 2x + 10 - 4 / (x + 1).