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what is the GCF of 8,22, and 6 ?

Oh wait i think i get it hold on...
8= 2^3
22=2*11
6=2*3

is that right so far??

notice how 2 is a factor in each, so we can factor out 2
8x^2+22x-6 = 2(4x^2+11x - 3)

how would we factor 4x^2+11x - 3 ?

Wouldn't you start simplifying the problem?

put another way:
what two numbers multiply to -12 and add to 11?

I'm not combining like terms

I'm asking for the two numbers because it will help factor

oh. Okay so 11 is a prime number so its only factor is one.

-12 could be 6,2,3,4

The first one

yep
-1*12 = -12
-1 + 12 = 11

so we can break up the 11x into -1x + 12x

Oh okay...

4x^2+11x - 3
4x^2-1x + 12x - 3

then we'd factor 4x^2-1x + 12x - 3 by grouping

group up the terms in pairs
4x^2-1x + 12x - 3
(4x^2-1x) + (12x - 3)
do you see what step is next?

Simplify them.

specifically you factor each group

how can you factor 4x^2-1x ?

I don't know?

we can factor out x, agreed?
4x^2 - 1x = x(4x - 1)

oh yeah since they are like terms

how would you factor 12x - 3 ?

Wouldn't you just do 12 -3 and then add the x

what factors do 12 and 3 have in common?

12x and -3 aren't like terms

oh duh 3

yeah so 12x - 3 = 3(4x - 1)

4x^2-1x + 12x - 3
(4x^2-1x) + (12x - 3)
x(4x - 1) + 3(4x - 1)

what factor shows up twice?

one

what else

the 4x-1 shows up twice

we can factor that out
x(4x - 1) + 3(4x - 1)
will turn into
(x+3)(4x - 1)

so hopefully you see how 8x^2+22x-6 factors to 2(x+3)(4x - 1)

I think so

Okay