Subtracting rational expressions

Find the difference. Express the answer as a simplified rational expression, and state the domain. We have two rational expressions, and we're subtracting one from the other. Just like when we first learned to subtract fractions, or add fractions, we have to find a common denominator. The best way to find a common denominator, if were just dealing with regular numbers, or with algebraic expressions, is to factor them out, and make sure that our common denominator has all of the factors in it-- that'll ensure that it's divisible by the two denominators here. This guy right here is completely factored-- he's just a plus 2. This one over here, let's see if we can factor it: a squared plus 4a plus 4. Well, you see the pattern that 4 is 2 squared, 4 is 2 times 2, so a squared plus 4a plus 4 is a plus 2 times a plus 2, or a plus 2 squared. We could say it's a plus 2 times a plus 2-- that's what a squared plus 4a plus 4 is. This is obviously divisible by itself-- everything is divisible by itself, except, I guess, for 0, is divisible by itself, and it's also divisible by a plus 2, so this is the least common multiple of this expression, and that expression, and it could be a good common denominator. Let's set that up. This will be the same thing as being equal to this first term right here, a minus 2 over a plus 2, but we want the denominator now to be a plus 2 times a plus 2-- we wanted it to be a plus 2 squared. So, let's multiply this numerator and denominator by a plus 2, so its denominator is the same thing as this. Let's multiply both the numerator and the denominator by a plus 2. We're going to assume that a is not equal to negative 2, that would have made this undefined, and it would have also made this undefined. Throughout this whole thing, we're going to assume that a cannot be equal to negative 2. The domain is all real numbers, a can be any real number except for negative 2. So, the first term is that-- extend the line a little bit-- and then the second term doesn't change, because its denominator is already the common denominator. Minus a minus 3 over-- and we could write it either as a plus 2 times a plus 2, or as this thing over here. Let's write it in the factored form, because it'll make it easier to simplify later on: a plus 2 times a plus 2. And now, before we-- let's set this up like this-- now, before we add the numerators, it'll probably be a good idea to multiply this out right there, but let me write the denominator, we know what that is: it is a plus 2 times a plus 2. Now this numerator: if we have a minus 2 times a plus 2, we've seen that pattern before. We can multiply it out if you like, but we've seen it enough hopefully to recognize that this is going to be a squared minus 2 squared. This is going to be a squared minus 4. You can multiply it out, and the middle terms cancel out-- the negative 2 times a cancels out the a times 2, and you're just left with a squared minus 4-- that's that over there. And then you have this: you have minus a minus 3, so let's be very careful here-- you're subtracting a minus 3, so you want to distribute the negative sign, or multiply both of these terms times negative 1. So you could put a minus a here, and then negative 3 is plus 3, so what does this simplify to? You have a squared minus a plus-- let's see, negative 4 plus 3 is negative 1, all of that over a plus 2 times a plus 2. We could write that as a plus 2 squared. Now, we might want to factor this numerator out more, to just make sure it doesn't contain a common factor with the denominator. The denominator is just 2a plus 2 is multiplied by themselves. And you can see from inspection a plus 2 will not be a factor in this top expression-- if it was, this number right here would be divisible by 2, it's not divisible by 2. So, a plus 2 is not one of the factors here, so there's not going to be any more simplification, even if we were able to factor this thing, and the numerator out. So we're done. We have simplified the rational expression, and the domain is for all a's, except for a cannot, or, all a's given that a does not equal negative 2-- all a's except for negative 2. And we are done.