 So let's consider, suppose we have a polynomial that factors as a product. Now if we expand this out, we get, and the thing that's worth noticing here is that the product of the coefficient of x squared and the constant will be pr times qs. And we'll rearrange this a little bit, that's ps times qr. And notice the x coefficients will be ps plus qr. And this suggests we could factor if we can find two numbers, ps and qr, where their product is the product of the constant and the coefficient of x squared, and their sum is a linear coefficient. And so we might put it this way, to factor ax squared plus bx plus c, find two numbers m and n, where m times n is ac, the product of the square and constant coefficients, and their sum is equal to b, the coefficient of the linear term. And then we'll rewrite bx as mx plus nx, then factor by grouping. So let's try to factor 3x squared plus 13x minus 10. So to factor something like this, we want to look for two numbers, m and n, where the product is equal to the product of the coefficient of x squared and the constant, and the sum is equal to the coefficient of x. So we want to find two numbers that multiply to 3 times minus 10, negative 30, and add to 13. Now as a general rule, the numbers that multiply to a given value are much more limited than the numbers that add to a given value. So let's find numbers that multiply to minus 30. Well, that's quite a few numbers, and so you might wonder which one's the right one. And here's the important thing to remember. The only way to determine if something is a factor is to check it. We don't know in advance which one of these will work. We have to check all of them until we find something that does, or determine that nothing works. Now remember, we're actually looking for two numbers that multiply to minus 30 and add to 13. So any of these multiply to minus 30, and so we look for a pair that adds to 13. So we could try our first pair, 1 and negative 30, and see if it adds to 13. Nope. We'll take a look at another pair, 2 and negative 15. Nope. However, because this adds to minus 13 and we want 13, this suggests that the negative of these two numbers, minus 2 and 15, will work. We check it out, and it does add to 13. So now we can rewrite our expression as 3x squared, 13x. Well, the reason this works is we can write 13 as minus 2 plus 15. So this 13x can be rewritten as minus 2x plus 15x. And now we're in a position to factor by grouping. So let's group our first two terms and our last two terms. Now our first two terms have a common factor of x, so we'll factor that out and get our second pair of terms has a common factor of 5, so we can remove that and factor as. And now both terms have a factor of 3x minus 2, so we'll remove that common factor, leaving us with x plus 5 times 3x minus 2. So let's try to factor something much more horrifying. So a useful idea to remember is to remove any common factors first. And if we look, we see that we can remove a common factor of 5. So let's factor that 5 out. So now to factor ax squared plus bx plus c, we want to find two numbers m and n where their product is a times c and their sum is b. And so we want to find two numbers that multiply to 12 times minus 12, negative 144, and add to minus 7 the coefficient of x. Again, there are fewer numbers that multiply to something than add to something, so let's find numbers that multiply to minus 144. And so our possibilities are, and of course it's obvious that the right one is, um, well, remember the only way to determine which one works is to try them out. Now if you're clever about this and think about it a little bit, you can go through this table of values efficiently. But I don't want to be clever. I want to go through this table of values one by one. So we'll try 1 and negative 144. We know they multiply to minus 144, and so the only question is do they add to minus 7? How about our next pair, 2 and negative 72? 3 and minus 48? Nope. 4 and negative 36? 6 and negative 24? 8 and negative 18? 9 and negative 16? Nope. Wait. No, that's right. So that means we should rewrite this minus 7x as 9x plus negative 16x. Actually to make the formatting a little bit easier, we'll rewrite that as minus 16x plus 9x. Everything else is still there. So we'll group the first two and the last two, and we'll look for a common factor in our first two terms. The common factor will be 4x, so we'll factor the first two terms. We can also factor our last two terms, and to now both of our terms have a 3x minus 4, so we'll factor out the 3x minus 4 to get our final factorization.