Question

Reading textbook example, how to factor this out? (Picture Provided).

Answer 1

How to factor this out? Yes, I can see the common factors, but still don't get what math was involved to get the answer.

Answer 2

Maybe consider a simpler example. Factor out the GCF in below expression
\[3(2x+1) + 7x(2x+1)\]

Answer 3

(2x+1)(7x+3)?

Answer 4

due to 2x+1 being a common factor, which leaves 3 and 7x remaining

Answer 5

Perfect! what about below :
\[3(2x+1)^2+7x(2x+1)\]

Answer 6

\[(2x+1)[3(2x+1)+7x\] ?

Answer 7

\[(2x+1)[3(2x+1)+7x]\]

yes! lets get back to the monster expression in original problem

Answer 8

ok

Answer 9

How many terms do you see ?

Answer 10

\[2(2x+1)^4(x^3-x+1)^3[2x+1)(3x^2-1)+5(x^3-x+1)]\] 3 terms

Answer 11

^ after taking the 3 common factors out

Answer 12

First notice that there are only two terms :

Answer 13

oh yes, 2 terms and then it is 3 common factors

Answer 14

Correct! your factorization looks good :
\[2(2x+1)^4(x^3-x+1)^3[(2x+1)(3x^2-1)+5(x^3-x+1)]\] 3 terms

Answer 15

just simplify the stuff inside square brackets on right side

Answer 16

\[(2x+1)(3x^2-1)=6x^3-2x+3x^2-1 + 5x^3-5x+5\]

Answer 17

and 5x^3 ... from +5(x^3-x+1)

Answer 18

\[11x^3+3x^2-7x+4\]

Answer 19

Hey wait, looks we forgot a "2" on left side factor

Answer 20

it should be
\[2(2x+1)^4(x^3-x+1)^3[\color{red}{2}(2x+1)(3x^2-1)+5(x^3-x+1)]\]

right ?

Answer 21

yea, that is the only part that is confusing, how does the 2 get in there? \[.... [2(2x+1)(3x^2-1) ....\]

Answer 22

other than that, I know how to factor out the "monster" expression