 So the thing to remember is that rational expressions are essentially fractions with polynomial numerators and denominators. And what that means is that every rule that we use for the arithmetic of fractions can also be applied to the arithmetic of rational expressions. So, for example, when we add or subtract fractions, as long as they have the same denominator, we keep the common denominator and add or subtract the numerators. So as long as our denominator polynomial isn't zero, if I'm going to add or subtract rational expressions, I'll add or subtract the numerators. Likewise, I can take any fraction and multiply numerator and denominator by the same number, and I can take a rational expression and multiply numerator and denominator by the same quantity. So, for example, let's add and maybe simplify if possible. So, since our denominators are the same, we can add the numerators, and that gets us x squared plus nine. We could try to simplify. So remember, a factor only matters if it's a common factor, and so the only simplification that's going to be possible is if our numerator has a factor of x minus three. And so we can check to see if x minus three is a factor of the numerator. So the question you've got to ask yourself is, is x squared plus nine equal to x minus three times something? If it is, the other factor must be x minus three. But if we expand out the right hand side, we see it doesn't work. So this means that x squared plus nine might factor, but any factorization we get won't be useful, so we'll leave it. How about a subtraction? Since the denominators are the same, we can subtract the numerators. Now we should be a little bit careful here. This is going to be x squared minus the quantity six x minus eight, which will be... Again, it's possible this may simplify, but we only need to check to see if x minus two is a factor of the numerator. So can the numerator be written as x minus two times something? Well the only possibility for the something is x minus four. We check it out, and we see that the factorization does work. And so our numerator is x minus two times x minus four. We can remove the common factor and be left with our final answer, x minus four. Or we can do this, and since our denominators are the same, we can add the numerators. Wait a minute, these are not actually the same. x minus nine is not identical to nine minus x. But here's a useful property to remember. Minus quantity a minus b is the same as b minus a. And what that means is that I can reverse the order of a subtraction by multiplying by negative one. So in our fraction, let's multiply numerator and denominator by negative one. That gives us negative ten as our numerator and reverses the order of the denominator. And now they have the same denominator. So we can add the numerators. We'll try to simplify. We'll see if x minus nine is a factor of the numerator. Can we write five x minus ten as x minus nine times something? And you should convince yourself that no matter what we put inside this parentheses, we will never be able to get five x minus ten. So even if the numerator factors, it's not going to be useful to find that factorization. And we might as well just leave our answer in this form.