Could someone explain why $\displaystyle 2x-10$ is not an irreducible in $\displaystyle Z[x]$ but is an irreducible in $\displaystyle Q[x]$.
Reducible means having a nontrivial factorisation. Irreducible means having no nontrivial factorisation. Nontrivial means that neither of the factors is a unit. If one of the factors is a unit then the factorisation doesn't count as a nontrivial factorisation. Since 2 is a unit in $\displaystyle \mathbb{Q}$, the factorisation 2x – 10 = 2(x – 5) doesn't count as nontrivial. That is why 2x – 10 is irreducible in $\displaystyle \mathbb{Q}[x]$.