Let $x,y\in {\mathbb{R}}^{n}$. Let A be a $n\times n$ positive-definite symmetric matrix. Is there a general formula for ${x}^{T}Ax\cdot {y}^{T}Ay$?

For example, let $x=\left[\begin{array}{c}2\\ 2\end{array}\right],\phantom{\rule{0.2cm}{0ex}}y=\left[\begin{array}{c}3\\ 3\end{array}\right],\phantom{\rule{0.2cm}{0ex}}A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Then ${x}^{T}Ax\cdot {y}^{T}Ay=8\cdot 18=144$

This is equal to $2(x\circ y{)}^{T}A(x\circ y)$ where $(x\circ y)=\left[\begin{array}{c}6\\ 6\end{array}\right]$ is the Hadamard product of x,y.

It seems that ${x}^{T}Ax\cdot {y}^{T}Ay=2(x\circ y{)}^{T}A(x\circ y)$ also works for $x=\left[\begin{array}{c}5\\ 5\end{array}\right],\phantom{\rule{0.2cm}{0ex}}y=\left[\begin{array}{c}7\\ 7\end{array}\right]$. Is this true in the general case, and if so, how do I prove it? If not, how can I find and prove a general formula?