To determine which system of equations best matches the direction field shown in the image, let's analyze the vector field behavior and the given answer options.

Step-by-Step Analysis:

1. Look at the behavior around $x = -1$ and $x = 2$:

   • The vector field seems to converge toward $(-1, 0)$, which suggests a critical point at $x = -1$.
   • Similarly, there's significant behavior around $x = 2$, likely another critical point.

2. Examine the general pattern:

   • The horizontal behavior of the arrows suggests a saddle point at the origin.
   • The arrows point towards the x-axis on both the left and right sides, and away from the y-axis as $y$ increases.

3. Check for symmetry:

   • The vertical behavior is consistent, with the arrows pointing up for positive $y$ and down for negative $y$, which indicates that $\frac{dy}{dt}$ could be a simple linear function of $y$, either $y+1$ or $y-1$.

Given Options:

1. First Option: $\frac{dx}{dt} = (x-1)(x+2), \quad \frac{dy}{dt} = -y+1$

   • The $(x-1)(x+2)$ factors suggest equilibrium points at $x = 1$ and $x = -2$. However, this doesn't match the observed critical points at $x = -1$ and $x = 2$. Also, the behavior for $\frac{dy}{dt} = -y+1$ suggests arrows pointing toward $y = 1$, which doesn't align well with the field.

2. Second Option: $\frac{dx}{dt} = (x+1)(x-2), \quad \frac{dy}{dt} = -y+1$

   • This suggests critical points at $x = -1$ and $x = 2$, which matches the direction field behavior better. The term $\frac{dy}{dt} = -y+1$ suggests that the arrows for $y = 1$ point towards $y = 1$ from below, which is consistent.

3. Third Option: $\frac{dx}{dt} = (x+1)(x+2), \quad \frac{dy}{dt} = y+1$

   • The $(x+1)(x+2)$ would indicate critical points at $x = -1$ and $x = -2$, which does not match the field at all.

4. Fourth Option: $\frac{dx}{dt} = (x-1)(x-2), \quad \frac{dy}{dt} = y+1$

   • This suggests critical points at $x = 1$ and $x = 2$, but it does not match the field pattern observed.

Conclusion:

The second option seems to match the critical points and behavior observed in the direction field: $\boxed{\frac{dx}{dt} = (x+1)(x-2) \quad \text{and} \quad \frac{dy}{dt} = -y+1}$

Would you like further details on how critical points affect direction fields?

Here are some related questions:

1. How do you analyze critical points in a direction field?
2. What are saddle points, and how are they recognized in vector fields?
3. How do equilibrium solutions affect the behavior of differential equations?
4. What is the significance of $\frac{dx}{dt}$ and $\frac{dy}{dt}$ in systems of differential equations?
5. How can you use nullclines to understand the direction of arrows in a field?

Tip: For analyzing direction fields, always identify the critical points first—they provide key insight into the behavior of the entire system!