To solve this, let's analyze the graph of the polynomial $p(x)$:

1. Roots of the Polynomial:

   • The graph touches the x-axis at $x = 2$ and bounces off, indicating that $(x - 2)$ is a factor with even multiplicity.
   • The graph crosses the x-axis at $x = 1$, indicating that $(x - 1)$ is a factor with odd multiplicity.

2. Behavior near Roots:

   • At $x = 2$, since the graph touches the axis and turns around, the multiplicity of $(x - 2)$ must be even.
   • At $x = 1$, since the graph crosses the axis, the multiplicity of $(x - 1)$ must be odd.

3. Possible Degree:

   • From the turning points and general shape, the total degree of the polynomial seems to be 5 (sum of the multiplicities of the factors).

Now let's evaluate the given choices:

• Option A: $p(x) = (2x - 1)^3 (x - 2)^2$

  • The factor $(2x - 1)^3$ gives a root at $x = 1$ with odd multiplicity (3), which is consistent.
  • The factor $(x - 2)^2$ gives a root at $x = 2$ with even multiplicity (2), which is consistent.

• Option B: $p(x) = (2x - 1)^2 (x - 2)^2$

  • Both factors have even multiplicity. Since the root at $x = 1$ requires an odd multiplicity, this is incorrect.

• Option C: $p(x) = (2x - 1)^3 (x - 2)^3$

  • Both factors have odd multiplicity. The root at $x = 2$ requires an even multiplicity, so this is incorrect.

• Option D: $p(x) = (2x - 1)^2 (x - 2)^3$

  • The factor $(2x - 1)^2$ gives an even multiplicity for the root at $x = 1$, which is incorrect.

Thus, the correct answer is Option A.

Would you like further details or explanations?

Here are 5 follow-up questions to consider:

1. How do multiplicities of factors affect the shape of a polynomial graph?
2. What happens when the multiplicity of a root is odd vs. even?
3. How do we determine the degree of a polynomial from its graph?
4. Why do turning points indicate changes in the sign of the derivative?
5. What is the role of leading coefficients in determining the end behavior of polynomials?

Tip: When analyzing polynomial graphs, always check how the curve behaves at the x-intercepts—whether it crosses or touches the axis—this will tell you the root multiplicity!