Simplify 𝑥 plus two over 𝑥 plus
three multiplied by two 𝑥 plus six over 𝑥 plus one.
So with a problem like this, we
could just multiply the numerators and multiply the denominators. However, what we always do when we
get a problem like this is look at them first to see, are there any section, so
numerators or denominators, that can be factored? Well, on inspection, we can see
that the numerator from the right-hand side can in fact be factored. So, what we’re gonna do is factor
this first. Well, what we can see is that with
our numerator on the right-hand side, we can take out two as a factor. When we do, we’ll have two
multiplied by 𝑥 plus three. So therefore, what we’ve got now in
our expression is 𝑥 plus two over 𝑥 plus three multiplied by two multiplied by 𝑥
plus three over 𝑥 plus one.
So, in these types of questions,
what we’re always looking to do is to divide through by any common factors. So, these common factors will
usually be apparent once we’ve done any factoring. And this is the case in this
question cause, as you can see, on the left-hand side, the denominator, we’ve got 𝑥
plus three. And now, on the numerator on the
right-hand side, we also have the factor 𝑥 plus three. So, what we now can do is divide
through by 𝑥 plus three. So now, using the standard rules
for multiplying fractions, what we’re gonna have is 𝑥 plus two multiplied by two
for our numerator, so that gives us two, multiplied by 𝑥 plus two. And then for our denominator, we’ve
got one multiplied by 𝑥 plus one, which just gives us 𝑥 plus one.
Now, we could leave it in this
form. But what we’re gonna do is one more
step before we give our answer. And that is to distribute across
our parentheses. And then if we distribute across
our parentheses, we have two multiplied by 𝑥 and two multiplied by two. So, therefore, we can say that if
we simplify 𝑥 plus two over 𝑥 plus three multiplied by two 𝑥 plus six over 𝑥
plus one, the result is two 𝑥 plus four over 𝑥 plus one.