Subjunctive mood - Discard B and C
D and E - the scientists believe , however, the structure of these two says it is widely believed.
IMO A

mv12 wrote:Scientists have recently received permission to research embryonic stem cells, derived from blastocysts or early-stage embryos, that they
believe to be capable of generating new cell growth and curing previously incurable ailments.
99
(A)that they believe to be capable
(B) that they believe are capable
(C)that they believe will be capable
(D)believed as capable
(E) believed to be capable

Skeleton of the sentence:
Scientists have recently received permission to research embryonic stem cells that they believe to be capable of generating...

"Believe X to be Y"is the correct idiom.

We need "that" after stem cells. if we don't use that after stem cells, we get--in the options D and E-- "stem cells, believed...."
Now, know the rule.
NOUN+COMMA+VERB-ED, here Verb-ed modifies the NOUN before COMMA. In the options D and E, the intended meaning of the sentence will change.
Answer is thus A.