 So what we've been doing so far is building up our arsenal, building up the different types of techniques we have to break down polynomials, to factor polynomials, right? So so far we've talked about when it comes to numbers, breaking down numbers into their prime factors, right? So we can break down numbers from the real number set into their prime factors and mainly we apply this to the rational numbers, right? So we can just break things down into their prime factors. We've talked about the GCF. We've talked about the GCF, the greatest common factor. So any multiple terms together, you know, add or subtract together, we look at the multiple terms and see what's common between them, we're being amounted, right? We've talked about the difference of squares. We've talked about the difference of squares where two things subtract from each other. We can factor them into the square root of one minus the square root of second square root of one plus the square root of second. And we've talked about simple trinomial factor and trinomials are of the following form, right? So trinomials are of the following form, ax squared plus bx plus c, where a cannot equal zero and for simple trinomials, a has to be equal to one, right? For simple trinomial factors, the ones we've done so far, a does not equal zero because if a equals zero, this would no longer be a quadratic equation, this would disappear, right? So we want the x squared term in there. And we've done the simple trinomial where a is equal to one. So basically what we have is of this form, it's x squared plus bx plus c. And these ones we know have a factor and we've already talked about, right? So what we're going to talk about right now is complex trinomial factoring where a is not equal to one. So a can't equal zero and a doesn't equal to one because if it equaled one, that's a simple trinomial factor. That's a simple trinomial factor, those things, right? So what we're going to start talking about now is factoring complex trinomials. And complex trinomials are the form or the following form where a doesn't equal zero and a doesn't equal one. So we actually end up with a number, a coefficient in front of the x squared, okay? The one thing you've got to be careful with is if there's a number in front of the x squared, what we should look for is the GCF first because the GCF, you always look for that first because if you can take out a greatest common factor, you know, if there's a number in front of the x squared and you can take the greatest common factor out and you can get x squared by itself just by the GCF method, then that just becomes a simple trinomial and we've already talked about those, right? So basically we're going to start talking about factoring trinomials where we can't take the a out just by taking out the GCF, okay? The method that I use to factor complex trinomials I think is referred to the four-step method or that's the name we came up with, right? In general, most teachers that I've encountered from, you know, from students from different schools is what they end up using, what they teach their students or most schools in the curriculum that in my area anyway, they teach them to do the, what's it called, the decomposition, okay? Now decomposition, I don't like it, I actually hate it, it's not a really an algorithm, right? Because what you have to do, the way it works is you split up the bx term into two terms and what you have to do is the first term that you're split it up with the ax squared and the other term you split it up with the c and then that's sort of a guessing part or a calculated guessing part, I guess, you know, the more you do the easier it becomes and then you factor out that way and it's not an algorithm, it's not, you know, it takes, you know, you have to take a step back and think about what you're going to do. I don't like that. The way I like doing, especially factoring with things that are just automatic, you're just breaking things down, I like doing them through an algorithm basically. I like doing them in a way where there's no thought process involved. I like to have the pen or the chalk in the wall or, you know, if you're doing a pen and paper, I like to have the pen and the paper do my thinking for me and I just like it to become a routine where you break things down and it's just, you know, automatic. You do the same thing all the time, okay? And the way I factor these is using the four step method. I'm not going to teach decomposition. If there's enough people that want decomposition part, you know, I'll come back to this and, you know, make a video to teach the decomposition, but what I'm going to do is teach the four step method. Now, one warning with the four step method. I've had students that, you know, go to their classrooms and the teacher has never seen the four step method. And I've had students where they write, you know, a complex trinomial quiz or an exam and because they use the four step method, the teacher, even though they get all the right answers, the teacher marks all the questions wrong because they didn't use decomposition. And, you know, I encountered this with a few students and I was quite upset about it, so I went online and I did a little search for a proof and I, you know, this is many moons ago and I, there was a proof that it came across where it showed how the four step method worked and it works all the time. I've never had it not work. And from the proof, if I recall correctly, you know, it proved that it works all the time. Now, if you're having problems in school, in the class that you're in, where your teacher doesn't like this method that I'm about to teach you, you know, drop me a line and I'll dig up the proof again or, you know, show them this video, you know, tell them how it works, where you learned it and it works all the time and it's a very quick algorithm. So, big warning for, you know, the complex trinomial information coming up. I haven't seen too many teachers or too many schools use it. So, make sure your teacher is okay with you using this process and if they're not okay with it using it, you know, get them to explain to you why they're not okay with it, okay? And if there's a good reason that they're not okay with it, then, you know, so be it. You're gonna have to use their method which is decomposition in general for what I've seen, okay? If they don't have a good reason that you can't use it, you know, get into a little discussion with them and see if they'll be okay if you end up using it. Okay? They might like it because it's, I find it to be simpler than the decomposition of it. So, let's go do a quick example, just a simple example and then from there we'll do a couple of more complicated examples.