Identify the position vector function.

The position of the particle as a function of time is given by:

$$\vec{r}(t) = [(5.0)t + (6.0)t^2]\hat{i} + [(7.0) - (3.0)t^3]\hat{j}$$

Substitute $t = 2.0$ s into the position function:

$$\vec{r}(2.0) = [(5.0)(2.0) + (6.0)(2.0)^2]\hat{i} + [(7.0) - (3.0)(2.0)^3]\hat{j}$$ $$\vec{r}(2.0) = [10.0 + (6.0)(4.0)]\hat{i} + [7.0 - (3.0)(8.0)]\hat{j}$$ $$\vec{r}(2.0) = [10.0 + 24.0]\hat{i} + [7.0 - 24.0]\hat{j}$$ $$\vec{r}(2.0) = (34.0\hat{i} - 17.0\hat{j})\, \text{m}$$

Substitute $t = 3.0$ s into the position function:

$$\vec{r}(3.0) = [(5.0)(3.0) + (6.0)(3.0)^2]\hat{i} + [(7.0) - (3.0)(3.0)^3]\hat{j}$$ $$\vec{r}(3.0) = [15.0 + (6.0)(9.0)]\hat{i} + [7.0 - (3.0)(27.0)]\hat{j}$$ $$\vec{r}(3.0) = [15.0 + 54.0]\hat{i} + [7.0 - 81.0]\hat{j}$$ $$\vec{r}(3.0) = (69.0\hat{i} - 74.0\hat{j})\, \text{m}$$

Displacement $\Delta \vec{r}$ is the change in position, given by $\Delta \vec{r} = \vec{r}(t_2) - \vec{r}(t_1)$.

$$\Delta \vec{r} = \vec{r}(3.0) - \vec{r}(2.0)$$ $$\Delta \vec{r} = (69.0\hat{i} - 74.0\hat{j}) - (34.0\hat{i} - 17.0\hat{j})$$ $$\Delta \vec{r} = (69.0 - 34.0)\hat{i} + (-74.0 - (-17.0))\hat{j}$$ $$\Delta \vec{r} = (35.0)\hat{i} + (-74.0 + 17.0)\hat{j}$$ $$\Delta \vec{r} = (35.0\hat{i} - 57.0\hat{j})\, \text{m}$$

The calculated displacement is $(35\hat{i} - 57\hat{j})$ m, which matches option (A).

The final answer is $\text{A) } (35\hat{i} - 57\hat{j})\, \text{m}$.