To determine the x-intercepts of the polynomial function $f(x) = -2x^2(x + 3)(x - 5)$, follow these steps:

Step 1: Set $f(x) = 0$

For x-intercepts, the y-coordinate of the function is zero, so: $f(x) = 0 \implies -2x^2(x + 3)(x - 5) = 0$

Step 2: Solve the equation

Using the zero-product property, each factor of the equation can independently equal zero:

1. $-2 \neq 0$ (ignore the constant $-2$ as it doesn't affect the solution).
2. $x^2 = 0$: This gives $x = 0$.
3. $x + 3 = 0$: Solving this gives $x = -3$.
4. $x - 5 = 0$: Solving this gives $x = 5$.

Step 3: Collect all x-intercepts

The x-intercepts are: $x = 0, \, x = -3, \, x = 5$

Final Answer:

The x-intercepts of the function are: $x = 0, \, x = -3, \, x = 5$

Would you like me to explain further, find additional properties (e.g., multiplicities), or address related concepts?

Here are 5 questions to expand your understanding:

1. What are the multiplicities of each x-intercept, and how do they affect the graph's behavior?
2. How does the negative leading coefficient ($-2$) influence the graph's end behavior?
3. What is the degree of this polynomial, and how many turning points might it have?
4. How does the x-intercept at $x = 0$ relate to the $x^2$ factor?
5. How would the graph differ if the constant $-2$ were positive instead?

Tip: Always check the degree of the polynomial to confirm the total number of x-intercepts, including multiplicities.