Note: More than one option maybe correct.
Alice and Bob are playing a game. At the beginning of the game, there is a cubic polynomial with integral coefficients written on the blackboard, which we denote as the starting polynomial. The two players take turns one by one. In each turn, the player either chooses any natural number $n$ and replaces the existing cubic polynomial $f(x)$ on the blackboard by any one of $f(n + x), f(nx), f(x) + n$ or just changes the sign of the coefficients of $x^2$ , i.e. if the existing polynomial is $a_0 + a_1x + a_2x^2 + a_3x^3$, then he can replace it by $a_0 + a_1x - a_2x^2 + a_3x^3$. Alice takes the first turn. Bob wins if, after finitely many moves, the cubic polynomial on the blackboard has all coefficients (upto $x^3$) non-zero and equal. For which of the starting polynomials can Alice ensure that Bob does not win in finite number of moves?
d) none of the above
It is an art of problem-solving question and I am not getting any clue. Please help how to solve these problems.