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.
$f(-5)=2(-5)^3+6\cdot(-5)^2-8(-5)+2=-250+150+40+2=-58.$
$f(-4)=2(-4)^3+6\cdot(-4)^2-8(-4)+2=-128+96+32+2=2.$
Since $-58\lt0\lt2$, according to the Intermediate Value Theorem, there must be a $0$ in the given interval.