[Integer and Integer-Valued Polynomials]  [Examples and Counterexamples. Constructions]

Authors: Yashchenko I.Vv, Blyunov A.d.; Golenischeva-Kumysova T.i.

For the quadratic equation $x^{2}+p x+q=0$, the coefficients $p$ and $q$ were increased by one. This operation was repeated four times. Provide an example of such an initial equation so that each of the five resulting equations has integer roots.
It is not difficult to check that the roots of the equation $x^{2}+3 x+2=0$ are -1 and -2. After increasing the coefficients by one, we get the equation $x^{2}+4 x+3=0$ with roots -1 and -3, then the equation with roots -1 and -4, followed by - with roots -1 and -5, and finally, the equation with roots -1 and -6.