Explanation

Blanks (i) and (ii) must describe what the governor wants to do to “the media’s hold on political communication.” From the first sentence, it is clear that the governor’s goal is to “excise” or eliminate the media as an intermediary between politicians and the public; this must be the meaning of the answer to Blank (ii), which describes the governor’s eventual, or long-term, goal. The only answer choice for Blank (ii) that has this meaning is “end,” so it is correct. The answer to Blank (i) must be a word that denotes the initial phase of a gradual process that ends with the complete elimination of the media’s influence; the answer choice that fits this meaning is “erode,” so it is correct. Thus the correct answer is erode (Choice B) and end (Choice F).
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The 1st sentence tells us that the governor wants to REMOVE (excise) the media from the politician-public relationship.
The last part of the 2nd sentence tells us that the governor is determined to (i) _________, and eventually (ii) _________, the media’s hold on political communication.
From this context, we might say that REMOVE fits nicely for blank ii (since that is the governor's goal)

We can also see that blank i is the first step towards eventually REMOVING the media's hold.
So, the word here should be a precursor to the actually removing the media's hold.
A word like LESSEN might work for blank i.

Now check the answer choices for words similar to LESSEN and REMOVE
The only ones that work are ERODE and END

Answer: B and F

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The governor has long been obsessed with excising the media from the politician-public relationship. That’s been the unifying aim of all her seemingly disconnected ventures since entering public life: a determination to (i) _________, and eventually (ii) _________, the media’s hold on political communication.

The governor has long been obsessed with cutting off the media from the politician-public relationship. Now what comes after the colon should elaborate or explain the same idea.

a determination to (i) cut, and eventually (ii) completely remove, the media's hold on political communication.

The word "eventually" suggests that I have to look at an extreme case of "cutting off".

The words from the choice list that make "the cut" are erode and end.
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Here the politician wants to excise the media from the relation between politicians and the public. Excise means to remove something. So all of the politicians policies aim to do this eventually. So we choose words accordingly, so regarding political communication, the politician's policies aim to 'erode' and 'end' the media's hold on political communication. So we choose B,F.