What is right? (English question)

Your question is ambiguous when you say "the answer indicates 'shows' in bold is not correct." This is ambiguous because you said (one line above) that the original text includes the word "shows."

The drug company profits from the experiment it shows that its product is effective.

This sentence is ungrammatical; it is made of two independent clauses that are not separated in any way. Therefore, without even analyzing its semantics, we know that the syntax of the sentence is incorrect. Before we correct it, we need to ensure that we preserve the intended meaning of the sentence. It should mean: "the drug company would benefit from an experiment that supports the efficacy of its drug." I purposefully changed the wording to ensure that I don't write the answer before I discuss it more.

Now, let's look at the first part of the sentence:

For example, if a drug company pays for an experiment to test how well one of its new products works, there is a special interest involved...

The important part is the word "if" (after "for example"). This indicates that we are working with a hypothetical situation. Now, we must reflect the hypothetical nature of the situation in the second part of the sentence by also adding an "if." The second half of the sentence would then read:

The drug company profits from the experiment if it shows that its product is effective.

Note: the "it" in the second half of the sentence was ungrammatical before I added "if," but now it is acceptable.

This is better, but we can still clean it up more; the sentence has "it" and "its" in it too many times. Instead, we can write it as follows:

The drug company profits if the experiment shows that its product is effective.

The verb tense of the verb "to profit" is currently the present indicative, but it should be conditional. "Shows" needs to be changed to its conditional form as well.

The drug company would profit if the experiment showed that its product is effective.

"Showed" isn't incorrect here, but a better word to use might be "proved." This lead us to our final answer:

The drug company would profit if the experiment proved that its product is effective.

There are other stylistic changes that can be made to further improve the sentence (in my opinion), but these would restructure the sentence, which would make the answer differ too greatly from the answer you were given.