You can use the given info to build a system and then solve for the coefficients.
The y-intercept is 9, therefore, x=0:
A root is -3:
Passes through (-2,3):
Passes through (1,16):
Now, you have 4 equations with 4 unknowns:
Solve the system and you get:
Therefore, the polynomial is:
This is one way to approach these. Perhaps try this on your other post regarding the quartic.