### Video Transcript

Expand and simplify two 𝑥 minus
six multiplied by three 𝑥 minus two multiplied by three 𝑥 plus two.

So in this question, what we need
to do is expand three brackets. Well, the way that we do it is to
actually expand two brackets first and then multiply everything inside the answer
from expanding two brackets by the other bracket. So what we’re gonna start with in
this question is three 𝑥 minus two multiplied by three 𝑥 plus two.

So the reason I’m gonna expand
these two first is because they are the difference of two squares. And what this means is that they
give a known answer or a known response when we expand the brackets. So I’m gonna expand them in the
normal way. But we’re gonna see what answer we
should get. So the result we’re expecting is
well if we’ve got 𝑎 minus 𝑏 multiplied by 𝑎 plus 𝑏, so in this case three 𝑥
minus two multiplied by three 𝑥 plus two where our 𝑎 is three 𝑥 and our 𝑏 is
two, then the result we should expect is 𝑎 squared minus 𝑏 squared. So in our case, that will be three
𝑥 all squared minus two all squared.

So if we’re gonna expand three 𝑥
minus two multiplied by three 𝑥 plus two, first of all, we need to multiply the
three 𝑥 in the first bracket by both terms in the second bracket. So we get three 𝑥 multiplied by
three 𝑥 which is nine 𝑥 squared. And then, we’ve got three 𝑥
multiplied by positive two which gives us positive six 𝑥. So we can see now that we’ve
multiplied our three 𝑥 by both of the terms in the second bracket.

So now we need to multiply the
negative two from the first bracket by both of the terms in the second bracket. So first of all, we have negative
two multiplied by three 𝑥 which gives us negative six 𝑥 and then negative two
multiplied by positive two which gives us negative four. So we’ve got nine 𝑥 squared plus
six 𝑥 minus six 𝑥 minus four. Well, now if we simplify, we can
see we’ve got plus six 𝑥 minus six 𝑥. So these will cancel, which gives
us nine 𝑥 squared minus four.

Now is this the result we were
expecting? Well, we were expecting 𝑎 squared
minus 𝑏 squared, so in our case three 𝑥 squared minus two squared. Well, if you square three 𝑥, you
get nine 𝑥 squared and if you square two, you get four. So this is the result we’re
expecting. Okay, so now we can rewrite our
expression. So now, we have two 𝑥 minus six
multiplied by nine 𝑥 squared minus four.

So what we’re gonna do now is
expand these two brackets. So we can use the same method as
before. So we’re gonna multiply the first
term in the first bracket by the first term in the second bracket. So we got two 𝑥 multiplied by nine
𝑥 squared which is gonna give us 18𝑥 cubed. And just reminding ourselves how we
actually did that, if you’re multiplying two 𝑥 by nine 𝑥 squared, first of all,
you multiply the coefficients, so two multiplied by nine. That gives us 18. Then, you’ve got 𝑥 multiplied by
𝑥 squared. So you add the powers. So we’re gonna get 𝑥 cubed.

Then next, we multiply the two 𝑥
from the first bracket by the negative four in the second bracket. So that gives us negative eight
𝑥. Then, we’ve got the second term in
the first bracket which is negative six multiplied by the first term in the second
bracket. So that’s gonna be negative six
multiplied by nine 𝑥 squared which gives us negative 54𝑥 squared. And then, finally, we’ve got plus
24. That’s cause you got negative six
multiplied by negative four. Negative multiplied by a negative
is a positive. So we got plus 24.

So what we’re gonna do now is
simplify. And to simplify, we can’t collect
any like terms because we don’t have any. And that’s because we’ve got
different powers of 𝑥 in each term. And you can’t add or subtract if
you got different powers of 𝑥. But what we can do is we can
rearrange it in descending powers of 𝑥. So therefore, we can say that if we
expand and simplify two 𝑥 minus six multiplied by three 𝑥 minus two multiplied by
three 𝑥 plus two, we get 18𝑥 cubed minus 54𝑥 squared minus eight 𝑥 plus 24.

So what I want to do now is check
our answer. And I’m going to do that so that I
can show you another method for expanding brackets that you might prefer to use. So we’re gonna check our answer by
expanding two 𝑥 minus six multiplied by nine 𝑥 squared minus four. And the method I’m gonna show you
is this one here, which is a grid method for expanding brackets.

So as you can see, I’ve got two 𝑥
and then minus six or negative six along the top. And then, I’ve got nine 𝑥 squared
and negative four down the left-hand side. So first of all, I multiply the
nine 𝑥 squared and the two 𝑥. So it gives us 18𝑥 cubed. And then, I multiply the negative
six and the nine 𝑥 squared which gives us negative 54𝑥 squared. Then, I’ve got the two 𝑥 and the
negative four, so that’s negative eight 𝑥, and then finally negative six and
negative four which gives us our 24.

And then, all we do is we write out
the terms in the grid. So we’ve got 18𝑥 cubed minus 54𝑥
squared minus eight 𝑥 plus 24. So that’s giving us the same answer
that we got using the other method. So we can say that we are happy
that 18𝑥 cubed minus 54𝑥 squared minus eight 𝑥 plus 24 is the correct
expansion.