Let \begin{equation}%\nonumber A:= \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{bmatrix} \end{equation} be a symmetric $3\times 3$ matrix with $a_{ij}=a_{ji}$ $(i,j=1,\dotsc,3)$. An observation $(x)(x)=(1)(x^2)$ leads to the fact that the following two statements are equivalent:
(1) $f(x):=\begin{bmatrix} 1 & x & x^2 \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{bmatrix} \begin{bmatrix} 1\\ x\\ x^2\\ \end{bmatrix} \ge0$ for $x\in\mathbb{R}$.
(2) For any $m\in\mathbb{R}$, $g(x,Y):=\begin{bmatrix} 1 & x & Y \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13}-\frac{m}{2}\\ a_{21} & a_{22}+m & a_{23}\\ a_{31}-\frac{m}{2} &a_{32} & a_{33}\\ \end{bmatrix} \begin{bmatrix} 1\\ x\\ Y\\ \end{bmatrix} \ge0$ for $x,Y\in\mathbb{R}$.
Does this implies that the following two problems are equivalent?
(1) Find all $a_{ij}$ $(i,j=1,2,3)$ such that
$\begin{bmatrix}
1 & x & x^2
\end{bmatrix}
\begin{bmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}\\
\end{bmatrix}
\begin{bmatrix}
1\\
x\\
x^2\\
\end{bmatrix}
\ge0$ for $x\in\mathbb{R}$.
(2) Find all $a_{ij}$ $(i,j=1,2,3)$ such that for any $m\in\mathbb{R}$, $$ \begin{bmatrix} 1 & x & Y \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13}-\frac{m}{2}\\ a_{21} & a_{22}+m & a_{23}\\ a_{31}-\frac{m}{2} &a_{32} & a_{33}\\ \end{bmatrix} \begin{bmatrix} 1\\ x\\ Y\\ \end{bmatrix} \ge0 $$ for $x,Y\in\mathbb{R}$.
Any reference, suggestion, idea, or comment is welcome. Thank you!