The problem provides the equation $y = x(x+3)(x-2)^2$ and its graph. The task is likely to analyze the function or graph but it is not specified. However, based on the equation given, the goal is to analyze the roots of the function.

1. Problem Description

The problem provides the equation $y = x(x+3)(x-2)^2$ and its graph. The task is likely to analyze the function or graph but it is not specified. However, based on the equation given, the goal is to analyze the roots of the function.

2. Solution Steps

The equation is $y = x(x+3)(x-2)^2$.

To find the roots (x-intercepts), we set $y = 0$ and solve for $x$:

$0 = x(x+3)(x-2)^2$

This equation is satisfied if any of the factors are equal to zero. Therefore, we have:

$x = 0$

$x+3 = 0 \implies x = -3$

$(x-2)^2 = 0 \implies x-2 = 0 \implies x = 2$

The roots are $x = 0$, $x = -3$, and $x = 2$. Note that $x = 2$ is a root with multiplicity

2. This means the graph touches the x-axis at $x = 2$ but doesn't cross it.

3. Final Answer

The roots of the equation $y = x(x+3)(x-2)^2$ are $x = -3, 0, 2$. The root $x=2$ has a multiplicity of 2.