Industrialization and modern methods of insect control have improved the standard of living around the globe
while at the same time they have introduced some 100,000 dangerous chemical pollutants, having gone virtually unregulated since they were developed more than 50 years ago.
Option Elimination -
"While" can be used to introduce participle phrases. "While" connects the "participle phrase" to the main clause of a sentence, indicating a simultaneous or contrasting action. Here, "simultaneous" action is important. When "While" is used to introduce a participle phrase, it indicates that the action described in the participle phrase is happening at the same time as the action in the main clause.
For example, While waiting for a bus, Sarah read a book. Or John Sang a song while playing a guitar. Or a contrasting action at the same time e.g. While cooking dinner, she listened to music.
Unlike "But," "while" can be used to introduce subordinating clauses or participle phrases.
(A) At the same time, they have introduced some 100,000 dangerous chemical pollutants, having the first "they" refer to "Industrialization and modern methods" while the 2nd "they" refers to "chemical pollutants." For "having," normally, the structure is "Having finished his work, John went for a walk" - the action of finishing work happened before John went for a walk. Present perfect participle is used to indicate an action that occurred before another past action (different from past perfect - in past perfect, the action has been completed in the past before another past action, but in present perfect participle, the effect of that action (that occurred before another past action) is present). Here, after "Having won," there is no other action.
Here, "having" while making sense with the "introduced some 100,000 dangerous chemical pollutants" doesn't make sense with "Industrialization and modern methods of insect control have improved the standard of living around the globe." So two issues, "they" and "having."
(B) while at the same time introducing some 100,000 dangerous chemical pollutants that have - - Here, the introduction of some 100,000 dangerous chemical pollutants has been happening at the same time Industrialization and modern methods of insect control have improved the standard of living around the globe. So the use of while is correct. The use of "that" is okay to introduce the essential information.
(C) while they have introduced some 100,000 dangerous chemical pollutants at the same time, which have - here there are a couple of issues. To show the simultaneous nature, we need a participle phrase. Moreover, to some extent, the use of "they" is ambiguous. They can refer to "Modern methods" or "Industrialization" and "Modern methods." Also, which can either refer to time or pollutants, though reference to pollutants makes more sense. Additionally, "At the same time" is a misplaced modifier because "Modern methods" or "Industrialization" didn't introduce the pollutants in one single moment just like that. The sentence means that as "Modern methods" or "Industrialization" improved the living standards, they also introduced chemical pollutants (in the same 50 to 100 years time).
(D) but introducing some 100,000 dangerous chemical pollutants at the same time that have - unlike "while" but needs a clause after it. Wrong.
(E) but at the same time introducing some 100,000 dangerous chemical pollutants, having - unlike "while" but needs a clause after it. Wrong.