p(x) has no zero between -1and +1, but does have a root between -2 and -1 at -1.1229 approx. That -2 is a lower bound is shown by using synthetic division:
-2 | -2 0 2 1 0 -2
| 4 -8 12 -26 52
-2 4 -6 13 -26 50
The alternating signs show that -2 is a lower bound. At x=0 p(0)=-2 and p(1)=-1. p remains negative as x is more positive. When x<-1.1229 p is positive.
Rational Zeros Theorem has 1 and 2 as the factors of the highest power of x and the constant term. These create the quotients +/-1 +/-1/2 +/-2. We already know that there appears to be but one zero p(-1.1229)=0.