0 + 0/2 + 0/3 + 0/4 + 0/5 = 0

boom

The question is whether p(x) has at least one real zero.

You gave a solution to a + b/2 + c/3 + d/4 + e/5 = 0, i.e. (0, 0, 0, 0, 0), but this is entirely unrelated. You need to show that there is a solution to p(x) = 0.

And you also need not come up with an explicit solution. You can just as well show that there is one, without explicitly saying what this solution is.