Answer
$f(-3)=-42$
$f(-2)=5$
Since $-42\lt0\lt5$, 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.
$f(-3)=3\cdot(-3)^3-10\cdot(-3)+9=-81+30+9=-42$
$f(-2)=3\cdot(-2)^3-10\cdot(-2)+9=-24+20+9=5$
Since $-42\lt0\lt5$, according to the Intermediate Value Theorem, the function must have a $0$ in the given interval.