Answer
$f(0)=-1$
$f(1)=10$
Since $-1\lt0\lt10$, 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.
$f(0)=8\cdot0^4-2\cdot0^2+5\cdot0-1=0-0+0-1=-1$
$f(1)=8\cdot1^4-2\cdot1^2+5\cdot1-1=8-2+5-1=10$
Since $-1\lt0\lt10$, according to the Intermediate Value Theorem, the function must have a $0$ in the given interval.