 So if the numerator or denominator of a rational expression is a quadratic, we can try to factor to simplify. And again, it's useful to remember that we can only remove common factors. So let's try to reduce x squared minus 3x minus 4 over 5x minus 20. So the thing to remember is that a factor only matters if it's a common factor. Since the denominator is linear, it'll be easier to factor. So we'll factor it first. 5x minus 20 is... So at this point we can try to factor the numerator. But remember the factorization of the numerator only matters if 5 or x minus 4 is a factor. Because a factor only matters if it's a common factor. And so we check. Can we write our numerator, x squared minus 3x minus 4, as x minus 4 times something? Well the only possibility for the something are first terms have to multiply out to x squared, and our constants have to multiply out to minus 4. And so our something has to be x plus 1. And so we check. Is it true that x squared minus 3x minus 4 is x minus 4 times x plus 1? And in this case we're fantastically lucky, and it is true. So equals means replaceable. So instead of x squared minus 3x minus 4, we can write x minus 4 times x plus 1. We have a common factor of x minus 4 that we can remove, leaving us with. Or let's take something like this. We can factor either the numerator or the denominator, but the denominator appears to be easier to factor, so we'll factor it first. We need two numbers that multiply to minus 12. Our possibilities are, but we don't know which one will work, so we'll have to try them out until we get something that multiplies to what we want. So we'll try how about x plus 1 times x minus 12, which isn't what we want. We'll try x plus 2 times x minus 6. Nope. x plus 3 times x minus 4, and that's exactly what we want. So our denominator factors. So remember, a factor only matters if it's a common factor. We want to factor the numerator, but just factoring in general isn't really worthwhile, unless either x minus 4 or x plus 3 is a factor. So we only need to check if x plus 3 or x minus 4 are factors of the numerator. So let's see if we can write 4x squared minus 19x plus 12 as x plus 3 times something. Now, in order to get 4x squared, our something has to have a 4x, and in order to get the plus 12, our something has to have a plus 4. And so we ask ourselves, self is 4x squared minus 19x plus 12 equal to x plus 3 times 4x plus 4. And the answer is, nope. But we do have that second possibility. We can see that if 4x squared minus 19x plus 12 is equal to x minus 4 times something. And so if we want to get a 4x squared, the something has to include a 4x. If we want to get a plus 12, our something has to include a minus 3. We check, and we see that this is the correct factorization. And so now we have a common factor of x minus 4, so we can remove our common factor and leave our final expression. So let's simplify. The numerator is horrifying. We don't want to try to factor that. So we'll try to factor the denominator because it looks at least like it's a little bit easier. In fact, 6x minus 8x squared factors as... Now looking a little bit ahead, we're used to seeing factors in the form something x plus or minus some number. So let's rearrange this 3 minus 4x using the fact that b minus a could be rewritten as minus a minus b. And so 3 minus 4x is the same as minus 4x minus 3. And we still have the 2x. Now when we go to factor the numerator, remember we only really care if we have a common factor. So we'll try to factor the numerator, and the only thing that matters is whether or not it has a factor of 4x minus 3. So if 12x squared plus 7x minus 12 can be factored as 4x minus 3 times something, it's got to be 3x plus 4. And since we live in a kind and gentle universe, the first thing we try as our factorization will always work. But I check it anyway, make sure that this really is the correct factorization. And it is, so we can factor our numerator, then remove our common factors.