f(x):=x^4+x^3-2x+1

case z=0
f(0)=1>0

lemma
for x>0, g(x):=xxx+xx+1/x > 2
∵using AM-GM, g(x) = xxx+xx+5*(1/5x) >= 7*(xxx*xx*(1/5x)^5)^(1/7) = 7/(5^(5/7))=2.21...

case x>0
f(x) = x*(g(x)-2) > 0

case x<0
y:=-1/x, then y>0
f(x) = (yyy+2yy+1/y-1)/yyy > (g(y)-1)/yyy > 0