  Let $p$ and $q$ be polynomials with degree $2$ over an arbitrary field
$\mathbb{F}$.
  In the first part of this article, we characterize the matrices that can be
decomposed as $A+B$ for some pair $(A,B)$ of square matrices such that $p(A)=0$
and $q(B)=0$. The case when both polynomials $p$ and $q$ are split was already
known. In the first half of this article, we complete the study by tackling the
case when at least one of the polynomials $p$ and $q$ is irreducible over
$\mathbb{F}$.
  In the second half of the article, we use a similar method to characterize,
under the assumption that $p(0)q(0) \neq 0$, the matrices that can be
decomposed as $AB$ for some pair $(A,B)$ of square matrices such that $p(A)=0$
and $q(B)=0$.
