The question in the image asks on which intervals the function $f(x) = (x - 2)^2(x + 3)$ is negative.

Step-by-step analysis:

1. Factoring the function: $f(x) = (x - 2)^2(x + 3)$

   • The term $(x - 2)^2$ is always non-negative because it's a square.
   • The term $(x + 3)$ can be positive, negative, or zero depending on the value of $x$.

2. Key points:

   • At $x = -3$, $(x + 3) = 0$, so $f(x) = 0$ at $x = -3$.
   • At $x = 2$, $(x - 2)^2 = 0$, so $f(x) = 0$ at $x = 2$.

3. Sign analysis:

   • For $x < -3$:
     • $(x + 3) < 0$, and $(x - 2)^2 > 0$, so $f(x) < 0$ (negative).
   • For $-3 < x < 2$:
     • $(x + 3) > 0$, but still $(x - 2)^2 > 0$, so $f(x) > 0$ (positive).
   • For $x > 2$:
     • $(x + 3) > 0$, and $(x - 2)^2 > 0$, so $f(x) > 0$ (positive).

Conclusion:

• The function is negative on the interval $(- \infty, -3)$ and zero at $x = -3$.
• The function is positive on the interval $(-3, 2)$, zero at $x = 2$, and positive thereafter.

Therefore, the correct intervals where the function is negative are:

• $(- \infty, -3)$

The selection of $(- \infty, -3)$ in the image is correct. The second interval $(-3, 2)$ is incorrect since the function is positive there.

Do you want further details or have any questions?

Here are some related questions you can explore:

1. What are the conditions for a function to be positive on a given interval?
2. How do you determine intervals of positivity or negativity using the sign chart?
3. How does the Intermediate Value Theorem guarantee the existence of roots?
4. What happens to the behavior of a function when its critical points are repeated roots?
5. How do we apply the concept of concavity to better understand the function's shape?

Tip: When working with polynomials, analyzing the sign of each factor and testing key points in different intervals can simplify determining where the function is positive or negative.