The subject is 'dating' so the verb has to be the plural 'suggests', which brings us to C, D and E.

The subjunctive mood states that 'suggest' should always be followed by 'that'. Hence C is the correct answer.
_________________

first of all "suggests" due to singular subject.


brings us down to c,d n e.

D can be ignored because of the use of past perfect...as there is no -ed form of verb as ref to it.
E can be ignored because of have i.e. present perfect.it suggests that it is still ice -free.We dont know that.
Suggests should always be connected with that as explained above.

So C is the best.

Radiocarbon dating of fossils taken from caves on islands along southeastern Alaska's coast suggest that at least a portion of the area was remaining ice-free during the last ice age.

a. suggest that at least a portion of the area was remaining There are 2 issues with this answer option:

i) What "suggests"? Radiocarbon dating. Therefore the subject of the verb "suggest" is singular, requiring the singular verb "suggests".
ii) The usage of the past continuous tense "was remaining" is problematic. Past continuous tense is used to emphasise the progress of an event in the past, often to overlap with a second event which is normally expressed in simple past tense (eg; Alan was cutting the grass the other day when the snake appeared). In this case, there is no second event, hence, it not clear why a past continuous tense is required. This is not a hard and fast rule (eg; "Alan was cutting the grass." is a perfectly acceptable sentence, although we can guess that there something else going on in the larger context, a second event), but a simple past expression is more concise and simpler.

b. suggest at least a portion of the area remaining Same subject-verb mismatch as explained in (i) above. Also, subjunctive mood "suggests" should be followed by "that".

c. suggests that at least a portion of the area remained Correct.

d. suggests at least a portion of the area that had remained Now here there is no ambiguity - the past perfect tense "had remained" necessitates a second event, since past perfect is used to show the order of happening of two events in the past. Incorrect. Also, subjunctive mood "suggests" should be followed by "that".

e. suggests at least a portion of the area to have remained. Two issues with this option:

i) Subjunctive mood "suggests" should be followed by "that".
ii) Present perfect "have remained" implies that the action (or its impact) continues till the present, which is an extrapolation of the sentence given in the question - a meaning shift.

Hope this helps.
_________________

a. suggest that at least a portion of the area was remaining --- wrong verb number
b. suggest at least a portion of the area remaining ----- wrong verb number
c. suggests that at least a portion of the area remained ---- CORRECT
d. suggests at least a portion of the area that had remained ----- Wrong meaning. "a portion" cannot be suggested, an action can be suggested. Therefore, we need "suggests THAT subject + verb.
e. suggests at least a portion of the area to have remained --- Wrong meaning. "a portion" cannot be suggested, an action can be suggested. Therefore, we need "suggests THAT subject + verb.
_________________

Radiocarbon dating of fossils taken from caves on islands along southeastern Alaska's coast suggest that at least a portion of the area was remaining ice-free during the last ice age.

a. suggest that at least a portion of the area was remaining
b. suggest at least a portion of the area remaining
c. suggests that at least a portion of the area remained
d. suggests at least a portion of the area that had remained
e. suggests at least a portion of the area to have remained.

'radiocarbon dating' should use a singular verb; hence, a and b are eliminated. D is not correct because perfect past tense 'had' implies that another event happened in the past. The usage of the tense of 'have remained' in E implies that the a portion of the area still remain till today, making no sense since we have the defined period 'during the ice age'

Radiocarbon dating of fossils taken from caves on islands along southeastern Alaska's coast suggest that at least a portion of the area was remaining ice-free during the last ice age.

a. suggest that at least a portion of the area was remaining What suggests? "Radiocarbon dating", singular. Therefore, we need singular form of the verb "suggests". Eliminate.

b. suggest at least a portion of the area remaining What suggests? "Radiocarbon dating", singular. Therefore, we need singular form of the verb "suggests". Eliminate.

c. suggests that at least a portion of the area remained Correct answer - above error is corrected and no new errors are introduced.

d. suggests at least a portion of the area that had remained Incorrect usage of past perfect "had remained" since it is not referring to the earlier of two actions in the past. Eliminate.

e. suggests at least a portion of the area to have remained. Incorrect usage of present perfect "have remained" since it is not referring to an action or effect continuing into the present. Eliminate.

Hope this helps.
_________________