The given function is $f(t) = -2t^9 - 4t^8 - 3t^3 + 2$.

To determine the long-run behavior as $t \to \infty$ or $t \to -\infty$, we focus on the term with the highest power of $t$, which dominates the growth of the function for large $|t|$. The highest power term in $f(t)$ is $-2t^9$.

Long-run Behavior:

1. As $t \to \infty$:

   • The dominant term, $-2t^9$, grows negatively because of the negative coefficient $-2$. Thus, $f(t) \to -\infty$.

2. As $t \to -\infty$:

   • When $t$ is negative, $t^9$ is also negative (since 9 is odd). Multiplying this by $-2$, the term $-2t^9$ becomes positive. Hence, $f(t) \to \infty$.

Final Answer:

• As $t \to \infty, f(t) \to -\infty$.
• As $t \to -\infty, f(t) \to \infty$.

Let me know if you would like further explanation or assistance! Here are some related questions for better understanding:

1. How do lower-order terms affect the behavior of a polynomial function in the long run?
2. Why does the sign of the leading coefficient influence the end behavior?
3. What happens if the highest power of $t$ is even instead of odd?
4. How do you analyze the long-run behavior of functions that are not polynomials (e.g., exponential or rational functions)?
5. Can we generalize this concept to multivariable polynomial functions?

Tip: Always start with the leading term when analyzing the long-run behavior of a polynomial!