Special products of binomials
Find the product 2x plus 8 times 2x minus 8. So we're multiplying two binomials. So you could use FOIL, you could just straight up use the distributive property here. But the whole point of this problem, I'm guessing, is to see whether you recognize a pattern here. This is of the form a plus b times a minus b, where here a is 2x and b is 8. We have 2x plus 8 and then 2x minus 8. a plus b, a minus b. What I want to do is I'm just going to multiply this out for us. And then just see what happens. Whenever you have this pattern, what the product actually looks like. So if you were to multiply this out, we can distribute the a plus b. We could distribute this whole thing. Distribute the whole a plus b on the a and then distribute it on the b. And I could have done this with this problem right here, and it would have taken us less time to just solve it. But I want to find out the general pattern here. So a plus b times a. So we have a times a plus b, that's this times this. And then a plus b times negative b, that's negative b times a plus b. So I've done distributive property once, now I could do it again. I can distribute the a onto the a and this b and it gives me a squared. a times a is a squared, plus a times b, which is ab. And now I can do it with the negative b. Negative b times a is negative ab or negative ba, same thing. And negative b times b is negative b squared. Now, what does this simplify to? Well, I have an ab, and then I'm subtracting an ab. So these two guys cancel out and I am just left with a squared minus b squared. So the general pattern, and this is a good one to just kind of know super fast, is that a plus b times a minus b is always going to be a squared minus b squared. So we have an a plus b times an a minus b. So this product is going to be a squared. So it's going to be 2x squared minus b squared minus 8 squared. 2x squared, that's the same thing as 2 squared times x squared, or 4x squared. And from that, we're subtracting 8 squared. So it's going to be 4x squared minus 64.