I am looking at this case where the Local-to-Global Principle does not hold. This is a problem in Katok's book, $p$-adic Analysis Compared with Real, (Exercise 46).
Show that $$(x^2-2)(x^2-17)(x^2-34)=0 \qquad (\star)$$ has a root in $\mathbb{R}$ and in the $p$-adic numbers $\mathbb{Q}_p$ for all primes $p$, but no solution in $\mathbb{Q}$.
One can quickly verify that that $(\star)$ has roots in $\mathbb{R}$ since $\pm\sqrt{2}$, $\pm\sqrt{17}$, and $\pm\sqrt{34}=\pm\sqrt{2}\sqrt{17}$ are irrational. Moreover, this also shows that $(\star)$ does not have a root in $\mathbb{Q}$.
How does one go about show that $(\star)$ has a root in $\mathbb{Q}_p$ for all primes $p$? Does one need to apply Hensel's Lemma here?