Your post title says it all!

The subjunctive mood must be used whenever you have an order or recommendation verb. Some examples of such verbs are order, mandate, recommend, ask and demand.

To properly use the subjunctive, the sentence structure is:

[subject] [verb] that [next subject] [infinitive form of next verb, without "to"]

For example:

Fred's boss demands that Fred be on time for work.
Financial experts recommend that investors do research.
Our professor asks that we be prepared for class.

On to the question!

When you have long complex sentences, you should start by stripping away the extraneous material. So, paraphrasing the original:



If we've trained ourselves well, we see the verb "ordered" and immediately think "subjunctive time!" There has to be a "that" after ordered - eliminate A, C and D.

Now we scan B and E and note that B has "should be measured". You never use "should" with an order or recommendation verb (you're already making an order or recommendation, so "should" is redundant) - eliminate B.

Only E fixes our subjunctive issues - choose E!

If we wanted to double check the structure of E:



There should actually be a "be" before published (which means that either the question has been mistyped or the original question isn't perfect), but otherwise all good.
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I do not agree with the explanation above..
first of all a verb like " order" can take both command subjunctive or infinitive form so you cannot eliminate options which do not have subjunctive indicators for ex if there were an option that started with " TO " so you cannot eliminate it ..

other verbs that fall in this category are - order, require,ask,beg,intend,urge etc

so coming back to options
Despite protests from some waste-disposal companies, state health officials have ordered the levels of bacteria in seawater at popular beaches to be measured and that the results be published.

The colored part is the descriptive phrase describing state health officials so subject is state health official (SHO)
a)SHO ordered the levels of bacteria ...????-doesnt make sense
b)being is always a red flag eliminate it there is a 1% chance that on gmat being is the correct option
c)k.sentence has to use either TO( form ) or THAT ( command subjunctive )
d)same problem as was with a - option doesn';t make sense
e) starts with THAT so that is correct:
THAT x and y
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From what I gathered out of the discussion, and as you rightly doubted, it appears that option E is missing be at the end.

(E) that the levels of bacteria in seawater at popular beaches be measured and the results be
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I also go with E and agree with the view that "be" is missing in E.