Factoring quadratics: negative common factor + grouping

we need to factor negative 12x squared minus 30 8f plus 22 so a good place to start is to see if is there any common factor for all three of these terms when you look at them they're all even and we don't like a negative number out here so let's divide everything or let's factor out a negative two so this expression right here is the same thing as negative two negative two times what's negative 12x squared divided by negative two it's positive 6f squared negative 38 divided by negative 2 is positive 19 so it'll be positive 19 F and then 22 divided by negative 22 sorry join 2 divided by negative 2 is negative 11 negative 11 so we've simplified it a bit we have the 6f squared plus 19 F minus 11 we'll just focus on that part right now and the best way to factor this thing since we don't have a 1 here is the coefficient on the F squared is to factor it by grouping so we need to look for two numbers whose products is 6 times negative 11 so two numbers so a times B needs to be equal to 6 times negative 11 or negative 66 and a plus B needs to be equal to 19 a plus B needs to be equal to 19 so let's try a few numbers here so let's see 22 22 I'm just thinking of numbers that are roughly 19 apart because our going to be of different signs so 22 and three I think will work right if we take 22 times negative 3 that is negative 66 and 22 plus negative 3 is equal to 19 and the way I kind of got pretty close to this number as well you know they're going to be a different signs so the positive versions of them have to be about 19 apart and that worked out 22 and negative 3 so now let's we can we can rewrite this 19 F right here as the sum of the sum of two of negative 3 F and 22 F so it's negative 3 F plus 22 F that's the same thing as 19 F I just kind of broke apart and of course we have the six F squared and we have the minus eleven here now you're probably saying hey Sal why did you put the 22 here in the negative three there why didn't you do it the other way around when you put the 22 and then the negative three there and my main motivation for doing it I like to put the negative three on the same side with the six because they have the common factor of the three I like to put the 22 with the negative eleven they have the same cop factor of 11 so that's why I decided to do it that way so now let's do the grouping and of course you can't forget this negative two that we have sitting out here the whole time so let me put that negative two out there but that'll just kind of hang out for awhile let's do some grouping so let's group these first two and then we're going to group and then we're going to group this let me get a nice color here and then we're going to group this second two that's almost an identical color let me do it in this purple color and then we can group that second two right there so these first two we could factor out a negative 3f so it's negative 3f times 6x squared divided by negative 3f is negative two F negative 2f and the negative 3f divided by negative 3f is just positive F actually a better way to start instead of factoring out a negative 3f let's just factor on 3f so we don't have a negative out here we could do it either way but if we just factor out a three f6f squared divided by 3 F is 2f and then negative 3f divided by 3 F is negative one so that's what that factors into and then that second part in that dark purple color we can factor out an 11 so we get factor out an 11 and if we factor that out 22 F divided by 11 is 2 F and negative 11 divided by 11 is negative 1 and of course once again you have that negative 2 hanging out there you have that negative 2 now inside the parenthesis inside the parenthesis we have two terms both of which have 2f minus 1 both of which have 2f minus 1 as a factor so we can factor that out this whole thing is just an exercise and in doing the reverse distributive property if you will so let's factor that out so you have to F minus one times this 3f times that 3f and then times that plus eleven times at plus eleven let me do that in the same shade of a lot of that same shade of eleven same shade of purple right over there and if you you know you can distribute if you like to F minus one times three F will give you this term 2f minus one times eleven will give you that term and we can't forget that we still have that negative two hanging out outside I want to change the colors on it we have the negative two hanging out that same negative two over there and we're done factoring it negative 12x squared minus 38 F plus 22 is negative 2 times 2 F minus 1 times 3 F plus 11