Answer
See below.
Work Step by Step
The Intermediate Value Theorem says that if a continuous (polynomials are always continuous) function on an interval [a,b] takes values $f(a)$ and $f(b)$ at the endpoints, then the function takes all values between $f(a)$ and $f(b)$ at some point of the interval.
Evaluate the function at the endpoints.
$f(-1)=(-1)^4+8\cdot(-1)^3-(-1)^2+2=1-8-1+2=-6.$
$f(0)=(0)^4+8\cdot(0)^3-(0)^2+2=0-0+0+2=2.$
Since $-6\lt0\lt2$, according to the Intermediate Value Theorem, there must be a $0$ in the given interval.