Let $\mathbb{F}$ be a field with char $\mathbb{F} \neq 2$. Let $A \in M_n(\mathbb{F})$. Does there exist symmetric matrices $B,C \in M_n(\mathbb{F})$ such that $A=BC$?
The answer is yes when $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. But how about other fields?
This is true over all fields, including those of characteristic two. Moreover, one of the two symmetric matrices can be taken to be nonsingular. Let $A=P^{-1}CP$ where $C$ be the Frobenius normal form (a.k.a. rational canonial form) of $A$. Suppose for the moment that $C=SC^TS^{-1}$ for some symmetric matrix $S$. Then \begin{align*} A &=P^{-1}CP\\ &=P^{-1}SC^TS^{-1}P\\ &=P^{-1}S(PAP^{-1})^TS^{-1}P\\ &=\underbrace{P^{-1}S(P^{-1})^T}_{S_1}\,A^T\,\underbrace{P^TS^{-1}P}_{S_1^{-1}}\\ &=S_1A^TS_1^{-1} \end{align*} for some symmetric matrix $S_1$. Therefore $A=S_1S_2$ where $S_2=A^TS_1^{-1}$ is also symmetric because $S_2^T=S_1^{-1}A=A^TS_1^{-1}=S_2$. Alternatively, you may write $A=H_1H_2$ where $H_1=S_1A^T$ is symmetric and $H_2=S_1^{-1}$ is symmetric and nonsingular.
So, it suffices to prove that $C=SC^TS^{-1}$ for some symmetric matrix $S_1$. In turn, by considering the problem blockwise, it suffices to consider the special case where $C$ is a companion matrix (with ones on the first subdiagonal) for a polynomial $f(x)=x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0$. In this case, one may take $$ S=\pmatrix{c_1&c_2&c_3&\cdots&c_{n-1}&1\\ c_2&c_3&\cdots&c_{n-1}&1\\ c_3&\vdots\\ \vdots&c_{n-1}&1\\ c_{n-1}&1\\ 1}. $$ Reference: Olga Taussky, The role of symmetric matrices in the study of general matrices, LAA 5:147-154 (1972). From the citations appeared in Taussky’s paper, apparently this result was first mentioned, if not proved, by Frobenius himself.