1. **State the problem:** Determine the end behavior of the polynomial function $$f(x) = x^2(x - 5)(x^2 + 2)$$ for large values of $$|x|$$. 2. **Recall the rule for end behavior:** The end behavior of a polynomial function is dominated by its leading term (the term with the highest power of $$x$$) because as $$|x| \to \infty$$, lower degree terms become insignificant. 3. **Find the leading term:** Expand the factors to find the highest degree term: $$f(x) = x^2(x - 5)(x^2 + 2)$$ First, multiply $$x - 5$$ and $$x^2 + 2$$: $$ (x - 5)(x^2 + 2) = x \cdot x^2 + x \cdot 2 - 5 \cdot x^2 - 5 \cdot 2 = x^3 + 2x - 5x^2 - 10 $$ 4. **Multiply by $$x^2$$:** $$f(x) = x^2(x^3 + 2x - 5x^2 - 10) = x^2 \cdot x^3 + x^2 \cdot 2x - x^2 \cdot 5x^2 - x^2 \cdot 10 = x^5 + 2x^3 - 5x^4 - 10x^2$$ 5. **Identify the leading term:** The term with the highest power of $$x$$ is $$x^5$$. 6. **Determine end behavior:** Since the leading term is $$x^5$$, which has an odd degree and a positive leading coefficient, the end behavior is: - As $$x \to +\infty$$, $$f(x) \to +\infty$$ - As $$x \to -\infty$$, $$f(x) \to -\infty$$ 7. **Final answer:** The graph of $$f$$ behaves like $$y = x^5$$ for large values of $$|x|$$.