Question 986182
Replace f(x) with 0, then factor


f(x)=x^3+6x^2+9x
0=x^3+6x^2+9x
x^3+6x^2+9x = 0
x(x^2+6x+9) = 0
x(x+3)(x+3) = 0


Then use the zero product property to break up x(x+3)(x+3) = 0 into these three equations


x=0 or x+3=0 or x+3=0


The first equation x=0 means we have a root of 0. This root is of multiplicity 1


The other two equations are exact duplicates of one another. They both lead to x=-3 being the other root of multiplicity 2.


So in the end, the answer is <font color="red">c. 0</font>


The graph confirms this. Notice how we have a double root at x = -3. It doesn't cross over the x axis as it only touches the x axis here. Contrast that with the root at x = 0 where it crosses through the x axis one time.


{{{ graph( 500, 500, -10, 10, -10, 10, 0,x^3+6x^2+9x) }}}