Binomial Expansion: Someone explain please?

You dropped a bit: it must say the coefficient of x is -24.

Let's take the general case here:

(a-b)^p

If p=2: a^2-2ab+b^2

If p=3: a^3-3a^2b+3ab^2-b^3

If p=4: a^4-4a^3b+6a^2b^2-4ab^3+b^4

and so forth.

Now, when we set a=1 and b=(3/2)x as in the specific case we're given, the x term is the second one, the one with just one factor of b in it. And that term always has the form

-p * a^(p-1) * b

regardless of what p happens to be. Furthermore,

a^(p-1) =1 for any value of p, because we're dealing with a case where a=1.

So we're looking for a term where the coefficient of x actually has a value of -24, and we know that term will have the value

-pb = -p * (3/2)x

because -(3/2)x is the term we represented by b in the general formula. So all we have to do to find p is solve an easy equation:

-p * (3/2)x = -24x

p * (3/2) = 24

p = (2/3) * 24 = 16

and if we were to expand

(1-(3/2)x)^16

we would find that the second term was -24x.