a sufficient and necessary condition for the non-negativity of a polynomial

Question

Find a sufficient and necessary condition for the non-negativity of $$ f(A,B,C):=a_1 A^2 B+a_2 A^2 C+a_3 A^4+a_4 A^3+a_5 A^2+a_6 A B+a_7 A C+a_8 A+a_9 B^2+a_{10}B C+a_{11} B+a_{12}C^2+a_{13}C+1\ge0, $$ for all $A,B,C\in\mathbb{R}$. By letting $A^2:=D$ be a new variable, it is easy to see $f$ becomes a quadratic form, say $g(A,B,C,D)$. Then there are several approaches to a sufficient and necessary condition for the non-negativity of $g$ for all $A,B,C,D\in\mathbb{R}$. However, this condition is not (?) a sufficient and necessary condition for the non-negativity of $f(A,B,C)$. I think this problem seems to be related to the Hilbert's 17th problem. Any reference, suggestion, idea, or comment is welcome. Thank you!