jmc

By the Integer Root Theorem, any integer root must divide 7. Thus, the possible values of the integer root are 1, 7, $-1,$ and $-7.$

We can plug in each integer root separately to see what $a$ is in each case. For $x=1,$ $$1+3+a+7=0,$$so $a=-11.$ For $x=7,$ $a=-71.$ For $x=-1,$ $a=9.$ For $x=-7,$ $a=-27.$

Thus, the possible values of $a$ are $\boxed{-71,-27,-11,9}.$