Find a polynomial of degree n that could have the given graph. Answers may vary
The graph bounces at -6, so it has a factor of (x + 6)2.
The graph passes through -3, so it has a factor of (x + 3)
The graph bounces through 0, so it has a factor of x2.
The graph has this type of equation: f(x) = a(x + 6)2(x + 3)(x2)
To see if a is positive or negative, check the end behavior:
-- if you put in a very large positive number, (x + 6)2(x + 3)(x2) will be positive (which means that the graph rises on the right-end
-- however, it drops on the right-end, so a must be negative.
Answer: f(x) = -(x + 6)2(x + 3)(x2)
Now, if you have to make certain that it hits all the other points, we would have to do a lot more work ...