Factor 4y squared plus 4y minus 15.
So we don't have a leading coefficient of 1 here.
And it also doesn't look like we can factor out a 4,
since 15 is not divisible by 4.
So what we're going to try to do, is factor this by grouping.
In the factor by grouping, we want
to look for two numbers whose product is
equal to the product of 4 and negative 15.
We explained it in other videos why this will work.
But let's just look for those two numbers.
So the product is going to be equal to 4 times negative 15,
which is negative 60.
And their sum needs to be equal to this 4 right over here.
So a plus b has got to be equal to 4.
And if we can figure out an a and b that satisfy this,
so we can break up this 4y into an ay plus a by.
And then we can factor by grouping.
Now the first clue here is that the product of the numbers
is negative.
So that means that a and b must have different signs.
And since a plus b is equal to 4,
we know that the-- and I want you
to think about this a little bit.
We know that the difference of their absolute values
must be equal to 4.
Let me just make that concrete, that little statement I just
told you.
If I were to say, well, let's just say that I have,
let's ignore this for a second.
Let me just take two simple numbers.
If I have negative 2 times 3.
Those two, the product is going to be equal to negative 6,
and then their sum, negative 2 plus 3 is equal to one.
Notice, their product is negative,
so they had to have different signs.
And their sum is 1 and they have different signs.
And one is also the difference of their absolute values.
The difference between the absolute value of 3
and the absolute value of 2 is one.
That's what I'm talking about.
So this gives us a clue of essentially
how far apart are the absolute values of the number.
So let's think about all of the factors of 60.
And then remind ourselves that we're
looking for numbers that are 4 apart,
numbers whose absolute values are 4 apart.
One has to be positive, and one has to be negative.
So the factors of 60, you could do 1 and 60,
you could do 2 and 30.
I'm going to run out of space doing it this way,
let me write it as a column.
So you have 1 and 60, you have 2 and 30, you have 3 and 20,
you have 4 and 15, you have 5 and 12, you have 6 and 10.
This looks pretty tantalizing since these are 4 apart.
And then that looks like we are about done.
7 doesn't go into it, 8 doesn't go into it, 9 doesn't go do it.
We're done.
Now already, we already talked about the difference
of the numbers, the absolute value of the difference
has to be 4, so 6 and 10 looks pretty tempting.
But let's just go through all of these numbers over here.
So the possible sums here, if we make 1 positive and 60
negative, then it's going to be a negative 59.
Remember, I'm making sure that we have different signs.
If this is positive, this could be positive or negative.
And this is negative or positive.
I wrote it upside down here, to show
that their signs are different.
So it's either going to be negative 59 if this is positive
and this is negative, or it's going to be positive 69 if this
is positive and this is negative.
So it's negative or positive 59, I should say.
This right over here is going to be negative or positive 28.
You see that the absolute value of, the difference
of the absolute values of these numbers keeps showing up.
Now we know that we want to go to 4, so let's
just cut to the chase and go to 6 and 10.
If we take positive-- and we want
them to add up to a positive 4.
So the larger number should be positive.
So if we take positive 10 and negative 6,
their sum is definitely going to be equal to 4,
and positive 10 times negative 6 is negative 60.
So let's break it up like that.
So what we could do is break up the 4 as a sum,
break up the 4y as a sum of a negative 6y and a 10y.
So let's do that.
So 4y, I'm going to write as negative 6y plus 10y.
If you add these two together, clearly you get 4y,
and then we have the rest of the problem here.
We have the 4y squared minus 15.
Now we're ready to factor by grouping.
Let's group these first two terms,
and we can because they have common factors here.
They're both divisible by 2 and y.
And then we're going to group these two guys, both of them
are divisible by 5.
So if we factor this first one over here,
let's factor out a 2y over here.
4y squared divided by 2y just gives us a 2y.
And then negative 6y divided by 2y is negative 3, right?
y divided by y is 1, negative 6 divided by 2 is negative 3.
Then let's factor this guy over here.
We can factor out a 5, so this is plus 5 times 2y minus 3.
And now lucky for us, we can factor out a 2y minus 3.
So we have 2y minus 3, times 2y, plus 5.
And we are done.