Answer
$f(-1)=-6$
$f(0)=2$
Since $-6\lt0\lt2$, according to the Intermediate Value Theorem, the function must have a $0$ in the given interval.
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.
$f(-1)=(-1)^4+8\cdot(-1)^3-(-1)^2+2=1-8+1+2=-6$
$f(0)=0^4+8\cdot0^3-0^2+2=0+0-0+2=2$
Since $-6\lt0\lt2$, according to the Intermediate Value Theorem, the function must have a $0$ in the given interval.