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We're asked to factor 4y squared plus 4y, minus 15.

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And whenever you have an expression like this, where

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you have a non-one coefficient on the y squared, or on the

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second degree term-- it could have been an x squared-- the

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best way to do this is by grouping.

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And to factor by grouping we need to look for two numbers

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whose product is equal to 4 times negative 15.

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So we're looking for two numbers whose product-- let's

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call those a and b-- is going to be equal to 4 times

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negative 15, or negative 60.

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And the sum of those two numbers, a plus b, needs to be

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equal to this 4 right there.

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So let's think about all the factors of negative 60, or 60.

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And we're looking for ones that are essentially 4 apart,

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because the numbers are going to be of different signs,

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because their product is negative, so when you take two

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numbers of different signs and you sum them, you kind of view

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it as the difference of their absolute values.

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If that confuses you, don't worry about it.

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But this tells you that the numbers, since they're going

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to be of different size, their absolute values are going to

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be roughly 4 apart.

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So we could try out things like 5 and 12, 5 and negative

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12, because one has to be negative.

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If you add these two you get negative 7, if you did

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negative 5 and 12 you'd get positive 7.

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They're just still too far apart.

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What if we tried 6 and negative 10?

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Then you get a negative 4, if you added these two.

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But we want a positive 4, so let's do negative 6 and 10.

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Negative 6 plus 10 is positive 4.

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So those will be our two numbers, negative 6 and

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positive 10.

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Now, what we want to do is we want to break up

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this middle term here.

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The whole point of figuring out the negative 6 and the 10

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is to break up the 4y into a negative 6y and a 10y.

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So let's do that.

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So this 4y can be rewritten as negative 6y plus 10y, right?

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Because if you add those you get 4y.

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And then the other sides of it, you have your 4y squared,

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your 4y squared and then you have your minus 15.

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All I did is expand this into these two numbers as being the

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coefficients on the y.

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If you add these, you get the 4y again.

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Now, this is where the grouping comes in.

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You group the term.

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Let me do it in a different color.

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So if I take these two guys, what can I factor out

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of those two guys?

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Well, there's a common factor, it looks like there's a common

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factor of 2y.

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So if we factor out 2y, we get 2y times 4y squared, divided

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by 2y is 2y.

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And then negative 6y divided by 2y is negative 3.

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So this group gets factored into 2y times 2y, minus 3.

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Now, let's look at this other group right here.

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This was the whole point about breaking it up like this.

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And in other videos I've explained why this works.

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Now here, the greatest common factor is a 5.

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So we can factor out a 5, so this is equal to plus 5 times

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10y, divided by 5 is 2y.

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Negative 15 divided by five is 3.

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And so we have 2y times 2y minus 3, plus 5

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times 2y minus 3.

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So now you have two terms, and 2y minus 3 is a

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common factor to both.

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So let's factor out a 2y minus 3, so this is equal to 2y

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minus 3, times 2y, times that 2y, plus that 5.

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There's no magic happening here, all I did is

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undistribute the 2y minus 3.

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I factored it out of both of these guys.

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I took it out of the parentheses.

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If I distribute it in, you'd get back to this expression.

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But we're done, we factored it.

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We factored it into two binomial expressions.

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4y squared plus 4y, minus 15 is 2y minus 3,

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times 2y plus 5.

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