2021 Fall AMC 12A Problem 15

Recall that the conjugate of the complex number w = a + bi, where a and b are real numbers and i = \sqrt{-1}, is the complex number \overline{w} = a - bi. For any complex number z, let f(z) = 4i\ \overline{z}. The polynomial P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1 has four complex roots: z_1, z_2, z_3, and z_4. Let Q(z) = z^4 + Az^3 + Bz^2 + Cz + D be the polynomial whose roots are f(z_1), f(z_2), f(z_3), and f(z_4), where the coefficients A, B, C, and D are complex numbers. What is B + D?

(\textbf{A})\: {-}304\qquad(\textbf{B}) \: {-}208\qquad(\textbf{C}) \: 12i\qquad(\textbf{D}) \: 208\qquad(\textbf{E}) \: 304

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.