The problem provided shows the polynomial function $P(x) = 3x(x + 1)^2(x - a)$, where $a$ is a real number, and the task is to determine which of the graphs (A, B, C, or D) correctly represents this function.

Key features to analyze in the polynomial:

1. Roots: The polynomial has roots at $x = 0$, $x = -1$, and $x = a$.

   • $x = 0$ is a simple root.
   • $x = -1$ is a double root because of the squared term $(x + 1)^2$.
   • $x = a$ is another simple root.

2. Behavior at each root:

   • At $x = -1$, the double root, the graph will touch the x-axis and bounce off because double roots create a parabolic-like behavior at the intercept.
   • At $x = 0$ and $x = a$, the graph will cross the x-axis since these are simple roots.

3. End behavior: Since the leading term when expanded would have a degree of 4 (highest power is $x^4$) and a positive leading coefficient, the end behavior of the graph will be $y \to +\infty$ as $x \to \pm\infty$.

Analyzing the graphs:

Looking at the options provided, graph D is the correct choice. Here's why:

• The graph in option D has a root at $x = -1$, where the graph touches the x-axis and bounces off, indicating the presence of a double root.
• There is a root at $x = 0$, where the graph crosses the x-axis.
• The end behavior of the graph matches a degree-4 polynomial (it rises to infinity on both ends).

The other graphs do not show the correct behavior at $x = -1$ or do not reflect the proper end behavior for a polynomial of degree 4.

Would you like further details or explanations?

Related questions:

1. What is the significance of a double root in a polynomial's graph?
2. How does the degree of a polynomial affect its end behavior?
3. How can you find the roots of a polynomial algebraically?
4. What is the relationship between the multiplicity of a root and the behavior of the graph at that root?
5. How would changing the value of $a$ affect the graph of $P(x)$?

Tip: Always check the behavior of a graph at the roots and the end behavior when matching polynomials to their graphs.