If a,b,c,d are integers such that both (-2a) * (−3c) * b and

Don't pay attention to quotients. They don't have any influence.

To make the expressions easier for understanding rewrite them in the following way:

Expression abc = positive
Expression abd = negative

In order to be positive and negative correspondingly 'abc' and 'abd' components have to have following set of signs:
abc abd
+++ ++-
+-- +-+
-+- -++
--+ ---

So, we can see that for each pair c and d always have different signs. Therefore for all possible options c*d must be negative.

Nice question
Since (-2a)*(-3c)*b> 0 => 6*a*b*c>0 = > a*b*c>0----1
d*(-2b)*(5a)>0 => -10*d*b*a>0- ==> d*b*a<0 ----2

Now let us look at the choices
A and B -- a*b and b*c . Well a*b can be positive in which case c>0 and a*b<0 in which case c<0 . The same thing applies to choice B and hence both are not correct choices
> 0
Let us look at D and E - a*d and a*c respectively. For reasons noted above if b<0 then a*c<0 but if b>0 then a*c >0. Also a*d for the same reason can be +ve and -ve.
Which brings us to C. At this stage we can check this answer and go.. But let us see what is going on

if c<0 then from inequality (1) a*b*c> 0 ==> that a*b <0 . From inequality (2) d*b*a<0 we see that if a*b<0 then d>0 ===> c*d<0
if c>0 then a*b>0 . From (2) d<0 ==> again c*d<0 so answer choice C!

A simple analysis will help you.

The given expressions are (ignoring co-efficients):
abc is positive.
-abd is positive.

Note that both have ab common so whatever sign that brings, will be same in both products. If c and d had the same signs, abc and abd would have the same sign so abc and -abd would have opposite signs.
So c and d must have opposite signs so that both abc and -abd have the same sign (positive).

Answer (C)

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.