Answer
See graph and explanations.
Work Step by Step
Step 1. Given $f(x)=-x^3-4x^2+11x+30$, test symmetry $f(-x)=x^3-4x^2-11x+30$, no symmetry.
Step 2. The leading term is $-x^3$ and we can find its end behavior as rise to the left and fall to the right.
Step 3. List possible rational zeros as $\pm1,\pm2,\pm3,\pm5,\pm6,\pm10,\pm15,\pm30 $. Use synthetic division as shown, we have $f(x)=(x-3)(-x^2-7x-10)=-(x-3)(x+2)(x+5)$ and the zeros are $x=-5,-2,3$
Step 4. The y-intercept can be found as $f(0)=30$
Step 5. Use test points as necessary, graph the function as shown in the figure.