The function provided is $y = x(x - 1)^3(x + 2)^2$. Let's analyze its graph by examining the key properties:

1. Roots: The function has roots at $x = 0$, $x = 1$, and $x = -2$.

   • At $x = 0$, the factor is linear ($x$), so the graph will cross the x-axis at $x = 0$.
   • At $x = 1$, the factor is cubic $(x - 1)^3$, so the graph will exhibit a "cubic" behavior, crossing the x-axis with a flattening near this point.
   • At $x = -2$, the factor is squared $(x + 2)^2$, so the graph will touch the x-axis and bounce off (since it is an even power).

2. End Behavior: The highest degree term is $x^6$ (since $x(x - 1)^3(x + 2)^2$ expands to a degree 6 polynomial), which means:

   • As $x \to \infty$, $y \to \infty$.
   • As $x \to -\infty$, $y \to \infty$ as well (because the leading coefficient is positive and the degree is even).

3. General Shape:

   • There is a bounce at $x = -2$,
   • A crossing at $x = 0$,
   • A cubic-type crossing at $x = 1$.

By comparing this behavior to the given graphs:

• Option A shows a bounce at $x = -2$, a crossing at $x = 0$, and another crossing at $x = 1$, matching the behavior of the function.

Thus, Option A is the correct answer.

Would you like any further details or explanations? Here are some related questions you might find helpful:

1. How does the degree of a polynomial affect the end behavior of its graph?
2. What is the significance of the multiplicity of roots in determining the graph's shape at those points?
3. How can you determine the turning points of a polynomial graph?
4. What are inflection points, and how are they related to cubic terms in a polynomial?
5. How do you find the local maxima and minima of polynomial functions?

Tip: When analyzing polynomials, always check the degree, root multiplicities, and end behavior to understand the overall graph structure.