Answer
$-2,-\frac{1}{2},\frac{-5\pm \sqrt {17}}{2}$
Work Step by Step
List all possible rational zeros $\pm1,\pm1/2,\pm2,\pm4$
Use synthetic division to test a few values and we can found two zeros $-2,-1/2$ as shown, so
$P(x)=(x+2)(2x+1)(x^2+5x+2)$
The zeros are $-2,-\frac{1}{2},\frac{-5\pm \sqrt {17}}{2}$
