 So what happens when we try to multiply two rational expressions? So it's useful to keep in mind that rational expressions are essentially fractions. When we multiply two fractions we multiply numerator by numerator and denominator by denominator. So a over b times c over d is a c over bd. We don't really change much when we multiply rational expressions. When we multiply two rational expressions, we multiply numerator by numerator and denominator by denominator. There's one important thing to keep in mind. It's easier to factor and remove common factors before you multiply. So let's try to multiply two rational expressions. Now because our end goal is to remove any common factor, the thing to remember is that a factor only matters if it's a common factor. We'll factor the easy things first. This 8x minus 16 looks easy to factor. Equals mean it's replaceable, so instead of 8x minus 16, we'll write 8 times x minus 2. Next, we might see if x plus 3 is a factor of either x squared plus 5x plus 6 or x squared plus 3x plus 4. So we see if x plus 3 is a factor of x squared plus 5x plus 6. Is it possible for us to write x squared plus 5x plus 6 as x plus 3 times something? Well, that something has to have an x because the first terms multiply it x squared and has to have a plus 2 since our constant terms have to multiply to 6. Now it's important to understand that we don't know whether this factors this way. We still have to check it out. So we need to multiply this out to see if we get x squared plus 5x plus 6. So checking it out, we see that we do in fact get our product. And so our x squared plus 5x plus 6 will factor as x plus 3 times x plus 2. And again, equals means replaceable. So instead of x squared plus 5x plus 6, we could write x plus 3 times x plus 2. So again, a factor only matters if it's a common factor. So we check to see if x minus 2 is a factor of the only thing that isn't factored yet, x squared plus 3x plus 4. And so can we write x squared plus 3x plus 4 as x minus 2 times something? Well, what does something have to be? Since we want to get an x squared, the something has to include an x. And since we want to get a plus 4, the constant must be minus 2. And since we live in a kind and gentle universe, the first thing we try must always work. And so x squared plus 3x plus 4 is certainly equal to x minus 2 times x minus 2. Eh, I check it. And we find that it isn't equal to x minus 2 times x minus 2. And since it isn't, it doesn't make a difference if we can factor x squared plus 3x plus 4. We might as well just leave it. Of course, it's still part of our problem, so we should rewrite it. So now we can multiply numerator by numerator and denominator by denominator. Remembering that factored form is best, so we won't actually multiply it out. We'll just run them together to indicate the product. We'll remove our common factors. And since factored form is best, we'll leave our answer in this form. Or how about this horrifying mess? So again, we'll want to factor everything possible, but let's start with the easy things. And since 25 minus x squared is the difference of squares, that's easy to factor. So we'll factor it. And it's in a strange form because we're used to putting the variables first. And so we'll use the fact that minus a minus b is the same as b minus a, and a plus b is equal to b plus a, to rewrite our factors in a more familiar form. 5 plus x, because of commutativity, we can rewrite that as x plus 5. 5 minus x, well that's the same thing as minus x minus 5. So remember, a factor only matters if it's a common factor. And so we might see if either x minus 5 or x plus 5 is a factor of either our two denominators. So let's see if it's a factor of x squared plus 2x minus 15 first. So is x squared plus 2x minus 15 equal to x minus 5 times something? Since we have to get an x squared, our something has to include an x. Since we have to get a minus 15, our something has to include a plus 3. And so we can see if x squared plus 2x minus 15 is x minus 5 times x plus 3. And it isn't. But maybe it's equal to x plus 5 times something, and there's something has to be. And we check it out, and it is. So we have our factorization of the denominator, x squared plus 2x minus 15. So now we have a new factor in the denominator, x minus 3. So we'll see if x minus 3 is a factor of x squared plus 5x plus 6. And so is x squared plus 5x plus 6 equal to x minus 3 times something? Again, since we need an x squared, our something has to include an x. And since we need a plus 6, our something must also include a minus 2. But we have to check out whether this factorization is correct. And so we check and find. Nope, it's hard to overemphasize how important it is to check to see whether a known factor is a factor of another polynomial. Because if we don't have a known factor, we have to use a lot of trial and error. So in this case, as it's possible, a factor of x squared plus 5x plus 6 might be a factor of x squared minus 2x minus 8. We'll need to factor at least one of them. And since we have no idea what the factors might be, we're going to have to use trial and error factorization. So we'll factor x squared minus 2x minus 8. The numbers that multiply to minus 8 are, and we have to try them out. We'll try 1 and negative 8. Nope. We'll try 2 and negative 4. And we get lucky, it only takes us two tries. Again, we could go through this trial and error factorization with x squared plus 5x plus 6. But a factor only matters if it's a common factor. So we'll see if x plus 2 or x minus 4 is a factor of x squared plus 5x plus 6. So we'll try with x plus 2 first. Can we write x squared plus 5x plus 6 as x plus 2 times something? Well, if we can, the something must be, and we check, and we see that this is a factorization. So now we have everything in factored form. So we can remove any common factors. And since factored form is best, we'll leave it in this form.