Perhaps if he did leave the gun behind, he wanted to stage a murder suicide. Put the car in a harder to locate location (at least not one anyone would be looking for this couple or their car), hope bodies aren't found for a while, put him in the back seat like he shot her from the back seat and them himself. The problem with that is people don't usually shot themselves in the back of the head. Maybe he thought it would just go away and police wouldn't look into it further.
I guess he couldn't leave them where it happened if it was in fact at his house. He had to move them, but why not put Matthew in the front seat.