Factoring with the distributive property

What I want to do is start with an expression like 4x plus 18 and see if we can rewrite this as the product of two expressions. Essentially, we're going to try to factor this. And the key here is to figure out are there any common factors to both 4x and 18? And we can factor that common factor out. We're essentially going to be reversing the distributive property. So for example, what is the largest number that is-- or I could really say the largest expression-- that is divisible into both 4x and 18? Well, 4x is divisible by 2, because we know that 4 is divisible by 2. And 18 is also divisible by 2, so we can rewrite 4x as being 2 times 2x. If you multiply that side, it's obviously going to be 4x. And then, we can write 18 as the same thing as 2 times 9. And now it might be clear that when you apply the distributive property, you'll usually end up with a step that looks something like this. Now we're just going to undistribute the two right over here. We're going to factor the two out. Let me actually just draw that. So we're going to factor the two out, and so this is going to be 2 times 2x plus 9. And if you were to-- wanted to multiply this out, it would be 2 times 2x plus 2 times 9. It would be exactly this, which you would simplify as this, right up here. So there we have it. We have written this as the product of two expressions, 2 times 2x plus 9. Let's do this again. So let's say that I have 12 plus-- let me think of something interesting-- 32x. Actually since we-- just to get a little bit of variety here, let's put a y here, 12 plus 32y. Well, what's the largest number that's divisible into both 12 and 32? 2 is clearly divisible into both, but so is 4. And let's see. It doesn't look like anything larger than 4 is divisible into both 12 and 32. The greatest common factor of 12 and 32 is 4, and y is only divisible into the second term, not into this first term right over here. So it looks like 4 is the greatest common factor. So we could rewrite each of these as a product of 4 and something else. So for example, 12, we can rewrite as 4 times 3. And 32, we can rewrite-- since it's going to be plus-- 4 times. Well if you divide 32y by 4, it's going to be 8y. And now once again, we can factor out the 4. So this is going to be 4 times 3 plus 8y. And once you do more and more examples of this, you're going to find that you can just do this stuff all at once. You can say hey, what's the largest number that's divisible into both of these? Well, it's 4, so let me factor a 4 out. 12 divided by 4 is 3. 32y divided by 4 is 8y.