Playing with Polynomial

$Alex$ and $Alice$ alternately replace the stars in $x^4+\bigstar x^3+\bigstar x^2+\bigstar x+\bigstar=0$ by integers of their choice.

$Alex$ wins if he get a integral root after fourth step. Otherwise $Alice$ wins. Who wins?

[ NOTE - $Alex$ will replace the first star, i.e. coefficient of $x^3$ and so on.]