 Ah, I can't believe it. I changed the sign from bus 3. Don't mind it, dude. I'm such a dangling. I gotta do that again. So what we had, one of the functions, one of the equations that we solved from the four-step of a complex trinomial was the following. One of the questions that we've already done using the complex trinomial factoring method, which is the four-step method, right? We solved this using the four-step method. What we're gonna do is solve this using the quadratic formula and using the let statement, right? The reason we're gonna use the let statement is because this isn't a quadratic equation, right? This is, you know, a quartic equation. So what we need to do, because to the power of four, right, is a quartic trinomial, right? Because there's three different expressions. So what we're gonna do is from series two, and again, you should know series two inside out right now, series two, and that's just exponents and radicals. And you should be able to manipulate radicals and exponents to your will, right? That's how good you should be at exponents and radicals right now because that's a huge chunk of the math that we're gonna get into, especially with functions, you know, log functions and exponential functions and decay functions and growth functions and any type of function, you know, we don't want it to stay in the base level here, right? We want to kick it up a notch and the first place we kick it up in mathematics, just simple numbers, is going to the powers, right? And radicals are just fractions in the power and exponents are just numbers in the power, right? So knowing our radicals and exponents or exponents in this case, we know that x to the power of four can be written as x squared to the power of two. So we can rewrite this equation in the following form. We can write this equation in the following form where we broke down x to the power of four to x squared squared. And from this, what we can do is use the let statement to rewrite this in the form of a, in a simpler form of the quadratic equation so we can use the quadratic formula, right? Right now, this is basically a quadratic equation with another function, a squared function, a quadratic really embedded within the quadratic, right? But what we're going to do is simplify this a little bit further because what's going to happen is later on we're going to get way more complicated equations and we don't want to continue, you know, to keep complicated stuff inside the brackets and let that equal to the quadratic formula, right? So what we're going to do is say let w equal x squared. Whenever you're using the let statement, write it down because when you get to the bottom, whatever your letter was, if you use w, you should be able to look further up and see what w is equal to, right? You're sort of creating your own universe when you say let w equal x squared. You're sort of using your own parameters right now, right? So you get a favorite of control with the let statement in mathematics because, you know, you can again manipulate things to your liking, to your will, right? So what we can do is rewrite this original equation as and this is obviously a quadratic equation. So what we're going to do is, you know, write it as w, not x is equal to the quadratic formula but w is equal to the quadratic formula because the x was just a generic x, right? The x is equal to negative b plus or minus a square to b squared minus 4 is c over 2a. It's a generic x. That x means anything can equal that, right? So we're going to say w equals that and w is really x squared. But we're going to stick with w for now and then when we get to the end, when we get w equals whatever the numbers are, then we're going to do another substitution and let x squared equal those numbers. So what we're going to do is use the quadratic formula and just write it out. So w is equal to negative b, so that's negative 3. Again, the sign in front of the number always goes with the number, right? So this is negative 3, that's negative 5 and that's positive 2, right? So what we have is negative negative 3 is just positive 3. So it's going to be 3 plus or minus. Negative 3 squared is going to be 9 minus 4 times negative 5 times 2. All of it divided by 2a, which is 2 times negative 5. So all we've got to do is just simplify this right now, right? So we're going to have... So w is going to be equal to 3 plus or minus 7 over negative 10, right? So 3 plus 7 is going to be... 3 plus 7 is 10 divided by negative 10 is negative 1. So w is going to be equal to negative 1, right? And 3 minus 7 is negative 4 divided by negative 10 is going to be 4 over 10, which is 2 over 5. So w is going to be equal to negative 1 and w is going to be equal to 2 over 5, right? It's just breaking this down. Now, we haven't solved our original question yet. What we did, we got the answers for this expression here, right? This equation here. What we need to do is go back and resubstitute x squared for w and solve for x because your original question is going to be solved for x. Because x is what we had in the original question. We didn't have any w's here, right? W was something we introduced into the process to be able to simplify this and break it down to something that we can deal with that we can recognize, right? That we can use the quadratic formula for to be able to, you know, solve our original question. So what we need to do now is go back and resubstitute x squared into w. So what we're going to do is substitute x squared for w. So what's going to happen is this guy is going to replace this guy. So what we got is x squared is equal to negative 1 and x squared is equal to 2 over 5. And to solve this, all we got to do is square root both sides. So you take the square root of this side and take the square root of this side. Well, this is going to be x is equal to square root of negative 1. In the real number set, in the real number realm that we're working, this doesn't give us a solution, right? This is a complex number or imaginary number. So this guy has no solution and that's a symbol for no solution, okay, or an empty set, I guess. And this one, we're going to take the square root of both sides. Square root of x squared is just going to be x, right? That's why we take the square root of both sides to get to x, right? And the square root of 2 over 5 is going to be plus or minus square root of 2 over 5. So plus and minus square root of 2 over 5. And that is our solution for, you know, our original expression, which was what was it? Which was negative 5 x to the power of 4 minus 3x squared plus 2 is equal to 0. And if we're solving for it, this is our answer, right? x is equal to positive square root of 2 over 5 and x is equal to negative square root of 2 over 5. Okay. If we're trying to find the factors, okay, so let's just do it on the side. If the question said, you know, find the factors of this, the factors of this would be in the following form. Let's say they ask you to factor this, not to solve it, right? What we would do is do all the steps that we did before and what we had ended up with was the following, right? We had x squared is equal to negative 1 and x squared is equal to 2 over 5. Right? Now, this is what x equal to, right? Or x squared equal to. This one we couldn't take down any further, but this one we could and this one gave us the following answer. x is equal to square root of 2 over square root of 5 and x is equal to negative square root of 2 over square root of 5, right? Which is what we had, right? x was equal to plus or minus square root of 2 over square root of 5. So what we can do with this is, hopefully you can see that, so what we have, this one we can't break down any further, so what we do is grab this and bring it over to the same side as x and make that side equal to zero. So over here we would have x squared plus 1 is equal to zero, right? Over here we would multiply the square root of 5 up, right? Cross multiply this up and bring the 2 over. So what we would have here is square root of 5x and the x is not under the square root symbol. Minus square root of 2 is equal to zero and over here we would do the same thing. So what we would have is square root of 5x plus square root of 2 is equal to zero and what we did was break this original expression down into three separate terms where we factored them as far as we can go, right? And that's the point when it comes to factoring something. When they ask you to factor something you're breaking them down, you're taking them down to as far as you can go and this is as far as we can go with it, right? And if this is as far as we can go all you need to do is to recreate the original expression is multiply these things together. So this guy completely factored would be this times this times this and the above guy factored would just be this guy and you would just leave it there if your original question was to factored this guy.