 So factoring is the hardest easy problem in mathematics. It's easy to describe what you want to do, write an expression as a product. But usually, it's very hard to find the actual factors. So to factor an expression, it helps to think about what we're trying to get. And so the thing we might notice is that if we expand a product of two binomials, we get a quadratic expression. So if we multiply x minus 3 times x plus 7 we get, or if we multiply 2x minus 1 by 4x plus 3 we get, and so if we have a quadratic expression, we might be able to factor it as a product of two binomials. So if we start with x squared plus 4x minus 21, we could rewrite it as the product. And if we start with 8x squared plus 2x minus 3, we can factor it as 2x minus 1 times 4x plus 3. And so here's the easiest possible cases of factorization. Factor something where you know the product of two things is equal to that same thing. And in a kind and gentle universe, all factorization problems would be like this. Unfortunately, we don't live in that universe. Again, factoring is extremely difficult, so we'll break it into two main cases. We'll start with monic polynomials where the coefficient of the leading term is 1. So x squared plus 4x minus 21 is monic. Remember that if we write a term but don't write a coefficient explicitly, we can assume there's an unwritten coefficient of 1. And then we have 8x squared plus 2x minus 3, and the coefficient of x squared is not 1, and so this is not monic. So let's try to deconstruct this product. If I multiply x plus a times x plus b, I get, and I'll collect these like terms, ax plus bx, and rewrite them as a plus bx. And so now let's compare our product to our factors. And if we do that, we see that the constant term of the product, ab, is the product of the constant terms in our factors, a and b. Moreover, the coefficient of x in our product, a plus b, is the sum of the constant terms in our factors. Again, a and b. And this suggests the following. To factor x squared plus px plus q, we want to find two numbers, a and b, where the product, ab, is the constant term q, and the sum, a plus b, is the coefficient of x. If these numbers exist, x squared plus px plus q is x plus a times x plus b. For example, let's factor, if possible, x squared plus 5x plus 6. And so our theorem says to factor an expression like this, we want to find two numbers, a and b, where the product, ab, is the constant term q, and the sum, a plus b, is the coefficient of x. And if these numbers exist, we'll get our factorization. And so we notice that 6, our constant term, is 2 times 3, while 5, the coefficient of x, is 2 plus 3. And so I have two numbers, 2 and 3, whose product is 6, the constant term, and whose sum is 5, the coefficient of x. And that means x squared plus 5x plus 6 factors as x plus 2 times x plus 3. Now, unless you're so lucky that factors just fall out of the sky, you'll need to do a lot of trial and error to find factors. And even if you do go through the trial and error process, there's no guarantee that they exist. It's important to remember factoring requires patience and persistence. There is no substitute, at least for now. As with everything, the more you learn, the more you can do, and the easier it is. Since the number of factors that multiply to a given number is fewer than the number of things that add to the given number, we can focus on finding factors of the constant term. Moreover, since factoring is usually the first step in a problem, you'll want to check your factorization. To do that, multiply the proposed factors and see if they work. So, for example, if we want to factor x squared plus 8x plus 12, so remember that x squared plus px plus q factors as x plus a times x plus b, where q, the constant term in our product, is equal to a times b, the constant terms of our factors. So, we want to form a list of things that multiply to 12. And so those numbers are. And it is possible for these factors to be negative numbers. So don't forget that for every pair of positive factors, we also have another pair of negative factors. Now, we could find the factorization by finding a pair that adds to 8, but that's a bad way to do it, and so we won't use that approach. Instead, we'll focus on what we really want to do. We want to find a pair of factors that gives us x squared plus 8x plus 12. And so we'll try them out one by one. So our first pair is 1 and 12, and so the question you've got to ask yourself is, is x squared plus 8x plus 12 the same as x plus 1 times x plus 12? And remember, the only way to determine if a factorization works is to check it. So we try 1 and 12 and check. We'll want to expand the right-hand side. Now, if you think about it, that expansion won't be too difficult. Remember, the x and the x have to be there because they multiply it to x squared. The 1 and the 12 came from the things that multiply it to 12. So we know that when we expand the right-hand side, it's got to have an x squared and it's got to have a plus 12. And so the only thing that we have to check out is what's the x term. Since the first thing we try always works, this is our factorization. Oh, wait, these aren't equal. And that means 1 and 12 don't work, and so they do not give us a factorization. As the old saying goes, if at first you don't succeed, give up and produce YouTube videos. Wait a minute, that's not right. Who mess with my Q cards? The thing to remember is that factoring requires patience and persistence. There is no substitute. So 1 and 12 don't work. There's a whole bunch of things that might work. So let's try 2 and 6 and check. Is x squared plus 8x plus 12 the same as x plus 2 times x plus 6? And we expand and find that it is. And so that gives us our factorization.