Expand and simplify negative two 𝑥 minus four 𝑦 times two 𝑥 minus 𝑦.
We’ll start by copying down the problem exactly as it was written. Our strategy here will be to distribute negative two 𝑥 and negative four 𝑦 across their factor two 𝑥 minus 𝑦. Like this, we’ll need to multiply negative two 𝑥 by two 𝑥 and by negative 𝑦 then we’ll need to multiply negative four 𝑦 times two 𝑥 and negative four 𝑦 times negative 𝑦. This will all fall under expand. And after we expand, then we can simplify.
Here we go. We’ll start with negative two 𝑥 times two 𝑥. Multiplying those together gives you negative four 𝑥 squared. Our next term will be negative two 𝑥 times negative 𝑦. From that, the product is two 𝑥𝑦. We move on and we need to multiply negative four 𝑦 times two 𝑥, which we could say that would equal negative eight 𝑦𝑥. And finally we’ll multiply negative four 𝑦 times negative 𝑦. This will give us positive 𝑦 squared. It also finishes off our expand step. So we’ll move on to simplification. And in this step we ask, can any of the terms in this problem be combined together? Are there any like terms?
Let’s look at our two terms in the middle. We have positive two 𝑥𝑦 and then negative eight 𝑦𝑥. Before we do that, we can bring down the negative four 𝑥 squared and the four 𝑦 squared. These are not like terms and they cannot be combined, so they’re already in their simplest form. Back to the two in the middle, can we do anything differently here? One thing we can do is we can switch the 𝑥 and the 𝑦 with our negative eight. Because we’re working with multiplication here, it doesn’t matter the order that we list the terms. Once we do that, it’s really easy to see that two 𝑥𝑦 and negative eight 𝑥𝑦 are like terms and they can be combined. Two 𝑥𝑦 minus eight 𝑥𝑦 equals negative six 𝑥𝑦.
Again, we’ve already determined that negative four 𝑥 squared and four 𝑦 squared can’t be combined with anything else, so those values need to be brought down. Be careful that you copy them exactly the way you found them. We’re working with negative four 𝑥 squared. We make sure we don’t forget that negative. Since nothing else can be combined together, that completes the simplification step. The expanded and simplified form of the problem given above is negative four 𝑥 squared minus six 𝑥𝑦 plus four 𝑦 squared.