。
The question is whether she can be trusted.
Literal
Problem [topic-は] she [subj-が] can-trust [whether-かどうか]-[という thing-ということ] [literary-である].
The pattern ~かどうかということ ("the matter of whether or not") wraps an embedded yes/no question into an even more nominalized form than ~かどうか alone. The added ということ makes the embedded question itself an explicit noun-like object of the main predication, giving the sentence a heavier, more formal feel. 信用できる is the potential form of 信用する ("trust").