MathNet 2019-part-05 — 184 problems (EN–VI) Question 175

Each of the trinomials $x^2 + a x + b$ and $x^2 + a x + b + 1$ has at least one real root; moreover, all real roots of those trinomials are integers. Prove that the trinomial $x^2 + a x + b + 2$ has no real roots.