Final answer:
The question requires analysis of the function f(x) = -4(x-1)²(x²-9), which involves identifying its roots and end behavior, based on the components of the quadratic and cubic terms in the polynomial equation.
Step-by-step explanation:
The given function is f(x) = -4(x-1)²(x²-9). To discuss the function, we must first address its components. The function is a polynomial equation depicted as the product of a quadratic term and a cubic term. The term (x-1)² represents a quadratic function, while (x²-9) simplifies to (x+3)(x-3), representing the difference of squares, which is a factor of the cubic term.
Given the function, we could analyze its properties, such as roots, end behavior, and turning points. The roots of the function, where the function crosses the x-axis, can be determined by setting the function equal to zero. The roots would be at x=1 from the quadratic term, and x=3 and x=-3 from the cubic term. The end behavior of the function is identified by the leading coefficient and the highest power of x.