Factoring quadratics: negative common factor + grouping

We need to factor negative 12f squared minus 38f, plus 22. So a good place to start is just to see if, is there any common factor for all three of these terms? When we 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 2. So this expression right here is the same thing as negative 2 times-- what's negative 12f squared divided by negative 2? It's positive 6f squared. Negative 38 divided by negative 2 is positive 19, so it'll be positive 19f. And then 22 divided by negative 22-- oh, sorry, 22 divided by negative 2 is negative 11. So we've simplified it a bit. We have the 6f squared plus 19f, 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 as the coefficient on the f squared, is to factor it by grouping. So we need to look for two numbers whose product 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. So let's try a few numbers here. So let's see, 22, I'm just thinking of numbers that are roughly 19 apart, because they're going to be of different signs. So 22 and 3, 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 is, well, you know, they're going to be of different signs, so the positive versions of them have to be about 19 apart, and that worked out. 22 and negative 3. So now we can rewrite this 19f right here as the sum of negative 3f and 22f. That's the same thing as 19f. I just kind of broke it apart. And, of course, we have the 6f squared and we have the minus 11 here. Now, you're probably saying, hey Sal, why did you put the 22 here and the negative 3 there? Why didn't you do it the other way around? Why didn't you put the 22 and then the negative 3 there? And my main motivation for doing it, I like to put the negative 3 on the same side with the 6 because they have the common factor of the 3. I like to put the 22 with the negative 11, they have the same common 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 2 that we have sitting out here the whole time. So let me put that negative 2 out there, but that'll just kind of hang out for awhile. But let's do some grouping. So let's group these first two. 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. Well, 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-- 6f squared divided by negative 3f is negative 2f. And then 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 out 3f, so we don't have a negative out here. We could do it either way. But if we just factor out a 3f, 6f squared divided by 3f is 2f. And then negative 3f divided by 3f is negative 1. So that's what that factors into. And then that second part, in that dark purple color, can factor out an 11. And if we factor that out, 22f divided by 11 is 2f, and negative 11 divided by 11 is negative 1. And, of course, once again, you have that negative 2 hanging out there. Now, inside the parentheses, we have two terms, both of which have 2f minus 1 as a factor. So we can factor that out. This whole thing is just an exercise in doing the reverse distributive property, if you will. So let's factor that out, so you have 2f minus 1, times this 3f, and then times that plus 11. Let me do that in the same shade of purple right over there. And you know, you can distribute it if you like. 2f minus 1 times 3f will give you this term, 2f f minus 1 times 11 will give you that term. And we can't forget that we still have that negative 2 hanging out outside. I want to change the colors on it. And we're done factoring it. Negative 12f squared minus 38f, plus 22 is negative 2 times 2f minus 1, times 3f plus 11.