Eidolon Battle: Alexander
Alexander is slow, but hits hard. From the start of battle wait until Hope gives everyone Protect and Shell, while Fang defends, drawing Alexander's attention. Once everyone is protected, switch to setup and let him have it. After he smacks you one, switch to and heal yourself, then return to and alternate as necessary. When the Gestalt gauge is full, press Square/X to be victorious.
You'll obtain the Alexander Eidolith Key Item after the battle.
Hope's ATB gauge will increase by one segment, bringing his total to 4 (or 5, if you already bought the ATB node in the Crystarium).