AMC 12 2025a Problem 19

Let $a$, $b$, and $c$ be the roots of the polynomial $x^3 + kx + 1$. What is the sum $$a^3b^2 + a^2b^3 + b^3c^2 + b^2c^3 + c^3a^2 + c^2a^3?$$

\textbf{(A)}-k \textbf{(B)}-k+1 \textbf{(C)}~1 \textbf{(D)}~k-1 \textbf{(E)}~k