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- [Voiceover] Pause the
video and try to add

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these two rational expressions.

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Okay, I'm assuming you've had a go at it.

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Now we can work through this together.

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So the first thing that you might have hit

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when you tried to do it,

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is you realized that they
have different denominators

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and it's hard to add fractions

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when they have different denominators.

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You need to rewrite them

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so that you have a common denominator.

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And the easiest way to
get a common denominator

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is you can just multiply
the two denominators,

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especially in case like this

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where they don't seem
to share any factors.

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Both of these are about
as factor as you can get

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and they don't share anything in common.

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And so let's set up a common denominator.

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So this is going to be equal to

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it's going to be equal to something

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let's see, it's going
to be equal to something

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over our common denominator.

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Let's make it

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let's make it 2x,

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I'm going to do this in another color.

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So we're going to make it 2x-3

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times 3x+1

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times 3x+1

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and then plus

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plus something else

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over 2x-3

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2x-3

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times 3x+1.

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Times 3x+1.

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And so to go from 2x,

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to go from just a 2x-3
here the denominator

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to a (2x-3)(3x+1)

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we multiply the denominator by 3x+1.

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So if we do that to the denominator,

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we don't want to change the value

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of the rational expression.

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We'd also have to do
that to the numerator.

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So the original numerator was 5x.

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I'll do that in blue color.

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So the original numerator was 5x

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and now we're going to
multiply it by the 3x+1,

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so times 3x+1.

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Notice I didn't change the
value of this expression.

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I multiplied it by 3x+1 over 3x+1,

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which is 1 as long as
3x+1 does not equal zero.

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So let's do the same thing over here.

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Over here I have a denominator of 3x+1,

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I multiplied it by 2x-3,

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so I would take my numerator,

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which is -4x²

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and I would also multiply it by 2x-3.

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2x-3.

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Let me put parentheses around this

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so it doesn't look like
I'm subtracting 4x².

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And so then I can rewrite
all of this business

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as being equal to,

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well, in the numerator,

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in the numerator I'm
going to have 5x times 3x

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which is 15x²

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5x times 1, which is + 5x,

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and then over here,

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let me do this in green,

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let's see, I could do -4x times 2x

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which would be -8x²

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and then -4x times -3

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which is +12x².

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Did I do that right?

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Negative...

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Oh, let me be very careful.

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- 4, my spider sense could tell
that I did something shady.

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In fact, if you want to pause
the video you could see,

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try to figure out what
I just did that's wrong.

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So -4x² times 2x

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is -8x to the third power.

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- 8x³

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and then -4x² times -3 is 12x²

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and then our entire denominator

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our entire denominator

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we have a common denominator now

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so we were able to just add everything

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is 2x-3

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2x-3

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times 3x+1

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times 3x+1

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and let's see, how can we simplify this?

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So this is all going to be equal to,

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let me draw,

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make sure we recognize
it's a rational expression,

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and so let's see, we can look at, we can,

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our highest degree term here is the -8x³

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so it's -8

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- 8x³

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and then we have a 15x²

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and we also have a 12x².

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We could add those two together

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to get a 27x²

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so we've already taken care of this,

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we've taken care, let me
do it in that green color,

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so we've taken of this,

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we've taken care of those two
and we're just left with a 5x,

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so + 5x

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and then all of that is over

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2x-3 times 3x+1

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3x+1

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and we are

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and we are all done.

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It doesn't seem like there's any easy way

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to simplify this further.

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You could factor out an
x out of the numerator

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but that's not going to cancel out

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with anything in the denominator

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and it looks like we are all done.