Who has a winning strategy for a finite number of moves, the first or the second player?

Question

Two players play the next game: They start with the polynomial $$2013x^2+2012x+2011$$and play by turns. Each player in his turn subtracts from the current polynomial one of the following polynomials: $x^2, x, x^2-x+1$ or $x^2+x-1$ of his choice. If after a move the resulting polynomial has an integer root, then the player who made the move loses. Who has a winning strategy for a finite number of moves, the first or the second player?