RU-ALLRUSSIAN 1984 Problem 13

Starting with the polynomial $\mathrm{x^2 + 10x + 20}$ , a move is to change the coefficient of $\mathrm{x}$ by 1 or to change the coefficient of $\mathrm{x^0}$ by 1 (but not both). After a series of moves the polynomial is changed to $\mathrm{x^2 + 20x + 10}$ . Is it true that at some intermediate point the polynomial had integer roots?