how do you verify that i is a solution to z^4 -3z^3 -3z¯3z -4=0 and then find all other solutions?
You substitute z = i into the equation and show that it works.
Then since all the coefficients of the equation are real, you know that z = -i is also a solution. So you know that (z - i)(z + i) = z^2 + 1 is a factor of z^4 - 3z^3 - 3z - 4.
So factorise z^4 - 3z^3 - 3z - 4 as two quadratics. Then solve when the other quadratic is equal to zero.