My best advice here is to read every answer choice out loud and see which one flows best and makes the most sense to you. the correct answer is A. every other answer choice places "written too quickly" in an awkwardly.
B. "many errors, written too quickly" -- the hind part of that sentence is confusing because you can't tell what was written too quickly. because the comma comes right after "errors," one might assume that the errors themselves were written too quickly, rather than the article. the subject is unclear so this answer choice is incorrect.
C. "written too quickly about the recent election" is the first clause in that sentence. it doesn't make sense. what was written too quickly? what was about the recent election? again, the reader wouldn't know how to apply the info given in the first part of the sentence to the last part.
D. "the article written too quickly" -- written is assumed to be the verb, here. the article written doesn't make sense. the article WAS written does. but still, the "written too quickly about the recent election" is a mess in the middle of the sentence because when the subject is introduced first, as "the article" was, it's difficult to find your way through the misplaced modifier to find the verb that applies to this subject.
A gives the modifier first, separating it from the rest of the sentence with a comma to show that "written too quickly" is a conditional of sorts. because it was written too quickly... it had many errors. it's more logical and it doesn't split your subject and verb up so awkwardly.
