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