I am very confused on how to start this question, I know that x=0 when i=1/5 and i=1 but not sure where to go from here, any help would be really appreciated.
"Find a polynomial P of a complex variable x that has zeros at exactly the following points: x=5i−1,x=3i−3.
All these zeros should be of first order. Be sure to expand all products in your answer; also expand products of the form (a+bi)⋅x if they occur."
Polynomials can be written in a form that shows its zeros directly, that is, if $x_1, x_2, \ldots, x_n$ are zeros of a polynomial $p$ and $a$ is the leading coefficient, $p(x) = a(x - x_1)(x - x_2)\ldots(x - x_n)$.
To get the the standard form you only have to multiply it all together.
So in your case, $x_1 = 5i - 1$ and $x_2 = 3i - 3$. You do not have any conditions on what the leading polynomial coefficient should be, so let it be $1$. That gives you $$p(x) = (x - (5i - 1))(x - (3i -3))= \\\ x^2 - x(3i -3) - (5i - 1)x + (5i - 1)(3i - 3) $$
You are instructed to expand those produsts... $$ p(x) = x^2 - 3ix + 3x - 5ix + x + 15i^2 - 15i - 3i + 3= \\\ x^2 - 8ix + 4x -18i -12$$.