The graph provided represents the polynomial function $f(x) = x^4 - 3x^3 + x^2 + 3x - 2$, and you are asked to solve the inequality $x^4 - 3x^3 + x^2 + 3x - 2 < 0$ by interpreting the graph.

Step-by-step explanation:

1. Determine when $f(x) < 0$:

   • The inequality $f(x) < 0$ is satisfied where the graph is below the x-axis.
   • From the graph, we observe that the curve dips below the x-axis between the x-intercepts at $x = -1$ and $x = 2$.

2. Identify the intervals:

   • The function is negative (below the x-axis) between the points $x = -1$ and $x = 2$.
   • At $x = -1$ and $x = 2$, the function touches the x-axis but is not below it at those exact points. Therefore, these points are excluded from the solution set.

3. Conclusion: The solution to $x^4 - 3x^3 + x^2 + 3x - 2 < 0$ is the open interval $(-1, 2)$.

Thus, the correct answer is: $\boxed{(-1, 2)}$

Would you like further details or explanations?

Related Questions:

1. How do you determine when a function is positive or negative from a graph?
2. What is the significance of x-intercepts in polynomial inequalities?
3. How can you use the factorization of polynomials to find intervals for inequalities?
4. What does it mean when the inequality involves $\leq$ instead of $<$?
5. How would the solution change if the inequality was $f(x) > 0$?

Tip:

Always check the behavior of the function at x-intercepts and ensure that you know whether to include or exclude them in inequality solutions.