jmc

First we isolate $x$ by subtracting 5 from both sides. This gives $$-4x>12.$$Dividing by $-4$ and remembering to reverse the inequality gives us $$x<-3.$$The greatest integer that solves this inequality is $\boxed{-4}$.

We can check this. If we substitute $-4$ into the inequality we get $$5-4(-4)>17$$or $$5+16>17.$$This is true. If we substitute $-3$ we get $$5+12>17,$$which is false.