20 !
a polynomial function is shown below: f(x) = x3 − 3x2 − 4x + 12
which graph best represents the function?

F(x) = x3 − 3x2 − 4x + 12    ( should be   f(x) = x^3 − 3x^2 − 4^x + 12 )

is a cubic function because its highest power of x is 3.  Because x^3 is positive, the graph starts in Quadrant III and ends in Quadrant I.  

The y-intercept is (0,12).

The greatest possible number of real roots is 3; the minimum number of real roots is 1.

Note that x=3 is a root of this function f(x) = x^3 − 3x^2 − 4^x + 12.  Thus, (3,0) is an x-intercept.  x=2 and x= -2 are also roots and lead to zero y-intercepts.

Can you now choose the correct graph?

If not, ask questions and I'll try to help you further.