1. **State the problem:** We are given the polynomial function $$f(x) = -5x^3 - 30x^2 + 200x$$ and want to analyze it. 2. **Identify the function type:** This is a cubic polynomial function with terms involving powers of $x$ up to 3. 3. **Factor the polynomial:** First, factor out the greatest common factor (GCF) from all terms: $$f(x) = -5x^3 - 30x^2 + 200x = -5x(x^2 + 6x - 40)$$ 4. **Factor the quadratic inside the parentheses:** We look for two numbers that multiply to $-40$ and add to $6$. These are $10$ and $-4$. $$x^2 + 6x - 40 = (x + 10)(x - 4)$$ 5. **Write the fully factored form:** $$f(x) = -5x(x + 10)(x - 4)$$ 6. **Find the roots (zeros) of the function:** Set each factor equal to zero: - $-5x = 0 \Rightarrow x = 0$ - $x + 10 = 0 \Rightarrow x = -10$ - $x - 4 = 0 \Rightarrow x = 4$ 7. **Interpretation:** The function crosses the x-axis at $x = -10$, $0$, and $4$. 8. **Summary:** The polynomial $$f(x) = -5x^3 - 30x^2 + 200x$$ factors as $$-5x(x + 10)(x - 4)$$ with roots at $x = -10$, $0$, and $4$. This helps understand the graph's intercepts and shape.