Give an example of a polynomial $p(x)\in\mathbb{R}[x]$ such that the quotient ring $\mathbb{R}[x]/(p(x))$ is not a product of fields.

Question

By the Chinese Remainder Theorem, if $p(x)$ is a product of distinct irreducible polynomials, then $\mathbb{R}[x]/(p(x))$ is a product of fields. So I was thinking about $p(x)=(x-1)^2$. However, I was struggling with proving that it is not a product of fields. I know that $\mathbb{R}[x]/((x-1)^2)$ has zero divisors, but a product of fields has zero divisors also. Does anyone know how to do this? Thanks.