The correct option is
A We asked that he listen carefully to the directions before starting the project.
The subjunctive mood is used in subordinate clauses and refers to unreal or improbable present or future situations.
The present subjunctive is used to express, urgency, importance, desirability, demand, order, suggestion or request.
Thus, according to the rules, the correct answer will be Option A. The other remaining options do not express present subjunctive mood.
Option B expresses past subjunctive mood.
Option C expresses imperative mood.
Option D expresses subjunctive mood.