I would substitute z = a + bi into the equation and see if you get 0. <<< There is no way to get zero when working with symbols.
If we substitute x = a + bi into the equation, then we get:
p = a^5 + 5i a^4 b - 10 a^3 b^2 - 10i a^2 b^3 + 5 a b^4 + i b^5 + 3 a^3 + 9i a^2 b - 9 a b^2 - 3i b^3 - 2a^2 - 4i a b + 2 b^2 + 10
Is the right-hand side of this equation equal to zero? It is, but only after substituting one of the four unknown pair of Real values for a and b that make it so.
Then substitute z* = a - bi also, and see if you also get 0.
If you don't get 0 for both, you should get the same value at least.
Again, you won't get any values. You won't even get the same expression.