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.
$f(0)=-2\cdot0^2-3\cdot0+8=-0-0+8=8.$
$f(4)=-2\cdot4^2-3\cdot4+8=-32-12+8=-36.$
Since $-36\lt0\lt8$, according to the Intermediate Value Theorem, there must be a $0$ in the given interval.