We include different cases first and examine them separately. First for integral domain $E$, apparently a quadraric polynomial $ax^2+bx+c$ is a square in $E[x]$ iff $a, b$ are squares in $E$ and $b^2=4ac$. My pirmary interests for this question is over non-integral domains so here is one example in $\mathbb Z/12\mathbb Z$: $x^2+4x+1$ is not a square in $\mathbb Z/12\mathbb Z[x]$ but has its discriminant $=0$ and both the first and third coefficient a square in $\mathbb Z/12\mathbb Z$.
I guess this question will not be trivial in a general ring $A$ so mainly I am looking for references to related theorems/books on this topic. And I expect the following questions being answered if it can:
$(i).$ When is a quadraric polynomial in $\mathbb Z/n\mathbb Z[x]$ a square?
$(ii).$ When is a polynomial $Ax^2+Bx+C\in\mathrm{Mat}_n(R)[x]$ a square, when $R$ is a commutative ring with identity?
Thank you in advance.