Might $a(x)$ be irreducible in $F[x]$ but reducible in $F[x]/\langle p(x)\rangle$, where $a(x)$ is not a multiple of $p(x)$?

$F[x]/\langle p(x)\rangle $ does not have to be a field. Could you name an example, preferably one for a field another for a non-field just so I can see the difference?

It was suggested by Arturo that I should re-edit this question since it was not clear enough:
Let F be a field, and say $a(x)∈F[x]$ be irreducible. If $p(x)$ is not constant and not equal to an associate of $a(x)$, and we let $R=F[x]/(p(x))$, and identify F with its image in R, can $a(y)$ be reducible in $R[y]$?