A scheme whose underlying space is the product of the underlying spaces of schemes

We know that the product of two spectral topological spaces is spectral.

• If $$X$$ is the underlying space of the scheme $$\mathrm{Spec}\,\mathbb{Z}[x]$$, what is a simple example of an affine scheme whose underlying space is $$X\times X$$?
• If $$X$$ is the underlying space of the scheme $$\mathrm{Spec}\,\mathbb{C}[x]$$, what is a simple example of an affine scheme whose underlying space is $$X\times X$$?
• btw, your link shows you could associate a scheme in principle, can you associate a scheme functorially? Kind of modified product of schemes? – user138661 May 6 at 13:26
• @schematic_boi Meta discussion that may be relevant: meta.mathoverflow.net/questions/4200/flood-of-new-users – Yemon Choi May 6 at 22:56