<s>[INST] $(x^2+1)(y^2+1)=z^2+1$ 
Prove that $$(x^2+1)(y^2+1)=z^2+1$$ has infinitely many solutions when $x,y,z \in \Bbb N$

I couldn't even find one solution for this equation.
Any ideas on how to prove it? 
Thanks in advance.
 [/INST] $$\boxed{\color{red}{(x^2+1)(4x^4+1)= (2x^3+x)^2+1}}$$

Some motivation for that "proof without words"  $$z^2 = x^2+y^2+x^2y^2$$
Let $y=xt$ then we have $$z^2 = x^2(1+t^2+x^2t^2)$$
so if we put $t=2x$ we get $$z^2 = x^2(1+4x^2+4x^4)=x^2(2x^2+1)^2$$
So we have triples $(x,2x^2,2x^3+x)$ where $x$ is arbitrary integer.
</s>