Rearrange the parts of the sentence in correct order.
"Do people take"
P - rejection of every candidate?
Q - the trouble to go and
R - vote only to register their

Let's find the logical link between the fragments starting from the introductory line:
"Do people take..." needs a noun phrase object. "take the trouble" is a common idiomatic expression. Thus, Q follows immediately.

"...the trouble to go and..." needs verbs to complete the action. "...go and vote only to register their..." fits perfectly. Thus, R follows Q.

"...register their..." requires a noun object that matches the context of voting. "...rejection of every candidate?" fits perfectly. Thus, P follows R.

The complete meaningful sentence is: "Do people take the trouble to go and vote only to register their rejection of every candidate?"
Therefore, the correct sequence is QRP.