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Multiply (3x+2) by (5x-7). So we are multiplying two binomials. I am actually going to show you

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two really equivalent ways of doing this. One that you might hear in a classroom and it is kind of a more mechanical

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memorizing way of doing it which might be faster but you really don't know what you are doing

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and then there is the one where you are essentially just applying something what you already know

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and kind of a logical way. So I will first do the memorizing way that you might be exposed to and they'll use something called FOIL.

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So let me write this down here. So you can immediately see that whenever someone gives you

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a new mnemonic to memorize, that you are doing something pretty mechanical. So FOIL literally stands for

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First Outside, let me write it this way.....F O I L where the F in FOIL stands for First, the O in FOIL stand for Outside,

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the I stands for Inside and then the L stands for Last. The reason why I don't like these things

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is that when you are 35 years old, you are not going to remember what FOIL stood for and

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then you are not going to remember how to multiply this binomial. But lets just apply FOIL.

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So First says just multiply the first terms in each of these binomials. So just multiply the 3x times the 5x.

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So (3x. 5x). The Outside part tells us to multiply the outside terms. So in this case, you have

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3x on the outside and you have -7 on the outside. So that is +3x(-7). The inside,

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well the inside terms here are 2 and 5x. So, (+2.5x) and then finally you have the last terms. You have the 2 and the -7.

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So the last terms are 2 times -7. 2(-7). So what you are essentially doing is just making sure

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that you are multipying each term by every other term here. What we are essentially doing is multiplying,

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doing the distributive property twice. We are multiplying the 3x times (5x-7). So 3x times (5x-7) is (3x . 5x) plus (3x - 7).

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And we are multiplying the 2 times (5x-7) to give us these terms.

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But anyway, lets just multiply these out just to get our answer. 3x times 5x is same thing as (3 times 5) ( x times x)

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which is the same thing as 15x square. You can just do this x to the first time to x to the first.

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You multiply the x to get x squared. 3 times 5 is 15. This term right here 3 times -7 is -21

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and then you have your x right over here. And then you have this term which is 2 times 5 which is 10 times x. So +10x.

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And then finally you have this term here in blue. 2 times -7 is -14. And we aren't done yet,

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we can simplify this a little bit. We have two like terms here. We have this...let me find a new color.

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We have 2 terms with a x to the first power or just an x term right over here. So we have -21 of something and you add 10

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or in another way, you have 10 of something and you subtract 21 of them, you are going to have -11 of that something.

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We put the other terms here, you have 15... 15x squared and then you have your -14 and we are done.

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Now I said I would show you another way to do it. I want to show you why the

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distributive property can get us here without having to memorize FOIL.

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So the distributive property tells us that if we 're... look if we are multipying something times an expression, you just have to multiply

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times every term in the expression. So we can distribute, we can distribute the 5x onto the 3...,

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or actually we could...well, let me view it this way... we could distribute the 5x-7, this whole thing onto the 3x+2.

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Let me just change the order since we are used to distributing something from the left.

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So this is the same thing as (5x-7)(3x+2). I just swapped the two expressions. And we can distribute this whole thing

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times each of these terms. Now what happens if I take (5x-7) times 3x? Well, thats just going

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to be 3x times (5x-7). So I have just distributed the 5x-7 times 3x and to that I am going to add 2 times 5x-7.

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I have just distributed the 5x-7 onto the 2.

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Now, you can do the distributive property again. We can distribute the

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3x onto the 5x. We can distribute the 3x onto the 5x.

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And we can distribute the 3x onto the -7.

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We can distribute the 2 onto the 5x, over here

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and we can distribute the 2 on that -7.

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Now if we do it like this what do we get ?

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3x times 5x, that's this right over here. If we do 3x times -7, that's this term right over here.

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If you do 2 times 5x, that's this term right over here. If you do 2 times -7, that is this term right

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over here. So we got the exact same result that we got with FOIL.

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Now, FOIL can be faster if you just wanted to do it and kind of skip to this step.

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I think its important that you know that this is how it actually works.

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Just in case you do forget this when you are 35 or 45 years old and you are faced with multiplying binomial,

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you just have to remember the distributive property.