The graph of $f(x)=-x^{4}+x^{3}+8 x^{2}-12 x$ is shown below. Use the graph to factor $f(x)$. A. $f(x)=x(x+3)(x-2)^{2}$ B. $f(x)=-x(x+3)(x-2)^{2}$ C. $f(x)=-x(x-2)(x+3)^{2}$ D. $f(x)=-x(x-3)(x+2)^{2}$

Step 1 ：The question is asking us to factor the function \(f(x)=-x^{4}+x^{3}+8 x^{2}-12 x\). The roots of the function are the x-values where the function crosses the x-axis. These roots are the solutions to the equation \(f(x) = 0\). The roots of the function are also the values of x that make each factor in the factored form of the function equal to zero. So, we can use the roots of the function to determine the correct factored form of the function.

Step 2 ：The roots of the function are -3, 0, and 2. The root 2 is repeated twice because the function touches the x-axis at x=2 but does not cross it. This means that the factor corresponding to the root 2 is squared in the factored form of the function.

Step 3 ：The roots -3, 0, and 2 correspond to the factors \((x+3)\), x, and \((x-2)\) respectively. Since the leading coefficient of the function is -1, the factored form of the function is \(-x(x+3)(x-2)^2\).

Step 4 ：Final Answer: \(\boxed{f(x)=-x(x+3)(x-2)^{2}}\)