 So one of the ways we have a factoring is known as factor by grouping, so we'll take a look at this approach. So we'll start out with the following. We'll try and factor this expression, and the thing to recognize here is that all factoring is based on the distributive property. And what that means is we want to find a common factor. So remember definitions are the whole of mathematics, all else's commentary, we want to find a common factor. A factor is something that is multiplied by something else. So here in this first term we have three things that are being multiplied, 12, x, and x plus 3. And in this second term we have two things that are being multiplied, 4, and x plus 3. And the thing to recognize is that both of these terms have x plus 3 as one of the things that are being multiplied. So both terms have a common factor of x plus 3, so we can remove it using the distributive property. It's helpful to remember that multiplication is commutative, so we typically express the distributive property using this A factor on the left, but we can reverse the order if the A factor is on the right, we can remove it as well. So we'll remove that common factor x plus 3. Remember that arithmetic is bookkeeping, keeping track of how much of what, and algebra is generalized arithmetic. So our x plus 3 factor has been taken care of. We had a 12x before, we still have it. We had a 4 before, and we still have it. And so our factorization becomes 12x plus 4 times x plus 3. Now if factoring problems like that just fell out of the sky, factoring would be easy. What makes it difficult is that the factoring problems don't drop out of the sky neatly grouped, we have to do something to get there. And so this leads to the idea of factor by grouping. And the idea is that we can group some of the terms of an expression. So in an expression like this, maybe I'll separate the terms into two groups. What goes in each group? I don't know, let's try it out. Let's go easy and put the first two terms into the first group, and the last two terms into the second group. If the universe is a kind and gentle place, we might be able to factor the terms in a group. So let's take a look at these two terms x to the third minus 3x squared. Now we note here that x to the third is x times x squared, while 3x squared is 3 times x squared. And that means x squared is a common factor, so we can remove that common factor. And since multiplication is commutative and it's traditional to put simpler factors first, we'll rewrite that as x squared times x minus 3. Similarly, this second set of terms, 3x minus 9, while 3x is really 3 times x, and 9 is really 3 times 3. And again, we see that both terms have this common factor of 3, so we can remove that common factor to get 3 times x minus 3. And now we look at our new expression, and we see that both of these terms have a factor of x minus 3. So we can remove that common factor. Arithmetic is bookkeeping. Algebra is generalized arithmetic. We had an x squared. We still have an x squared. We had a plus 3. We still have a plus 3. How about an expression like this? So again, we can try to group them, and we don't have to be too creative. Let's just try to group them first two terms, last two terms. And again, the important thing is we can try to factor the groups. So let's take a look at this first set of terms, x squared y plus xy. We can read this x squared y as xy times x. We could read xy as xy times 1. And both terms have a factor of xy, so we can remove it, leaving x plus 1. This other set of terms, 3x plus 3, both terms, 3x and 3, have a common factor of 3, so we can remove it, giving us a factorization. So now our new expression, both terms have a common factor of x plus 1, so we can remove it, giving a factorization. Or let's take another set of terms. Since grouping the first two terms and the last two terms worked in the other two cases, let's try to do the same thing this time. We'll group the first two terms, we'll group the last two terms. Let's take a closer look at those first two terms, 3x cubed plus 15x squared, and we see that there is a common factor of 3x squared. Removing that common factor gives us, for minus 4x minus 20, it can be a little tricky when you're dealing with a subtracted term. So a useful thing to remember is that you can rewrite any subtraction a minus b as a plus negative b. And so we can rewrite our expression minus 4x minus 20 as minus 4x plus negative 20. And this makes it a little bit easier to find that factor. Both of these have a common factor of minus 4, so we can remove that common factor of minus 4 and what's left over an x, a plus, and a 5. And so the first pair of terms factors, the second pair of terms factors, and they both have an x plus 5 as a common factor, so we can remove it. And what was there before is still there. We had a 3x squared, we still have a 3x squared, we had a plus negative 4, we have a plus negative 4, or we can use the fact that plus a negative is the same as a minus. We have a minus 4.