At < When does $R [x]/I $ has infinitely many idempotents? >, Er_Ro asked the following question.

Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotent and an unextended ideal $I $ in $R [x] $ such that $R[x]/I$ has infinitely many idempotent?

It is important and whenever $R $ is a semilocal ring, I want to know that is there any example in this case.