$2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$
Solution 1:
The bilinear map $\varphi:I\times I\to R$ given by $\varphi(u,v)=uv$ gives rise to a homomorphism $I\otimes I\to R$. If $2\otimes 2+X\otimes X=u\otimes v$ with $u,v\in I$, then $X^2+4=uv$. Assume that $u,v$ are monic of degree one, that is, $u(X)=X+2a$ and $v(X)=X+2b$. From $a+b=0$ and $ab=1$ we get a contradiction. If $\deg u=0$, then $u=\pm1$ which is not in $I$.