I need to show that for any $n$ there is an irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots.
As a hint I have that if $f(x) \in \mathbb{R}[x]$ is any polynomial having exactly $k$ distinct real roots, there exists $\epsilon > 0$ for which $f(x) +a$ has exactly $k$ real roots, for all $a\in \mathbb{R}$ with $|a|<\epsilon$.
Then, by starting with any polynomial $f(x) \in \mathbb Q[x]$ with exactly $n-2$ distinct real roots, and using the paragraph above, $f(x)+a$ has the same property for infinitely many $a\in\mathbb Q$. Now, make a judicious choice of $f(x) \in\mathbb Z[x]$ and $a \in\mathbb Q$ for which the Eisenstein irreducibility criterion can be applied.
If I understand the hints correctly, you are asked to explore $$ f(x)=(x-1)(x-2)...(x-(n-2))·(x^2+1)+ε $$ or, perhaps more suited for the modular explorations, $$ f(x)=(x^2+1)·(x-2)(x-4)(x-8)...(x-2^{n-2})+ε $$