To find the work done by a force during a displacement, we can use the formula:

$$W = \int_{x_1}^{x_2} \vec{F} \cdot d\vec{x}$$

Here, the force is given by $$\vec{F} = (2+3x) \hat{i}$$, and we need to find the work done during a displacement from $$x = 0$$ to $$x = 4 \mathrm{~m}$$. Since the force is only in the $$x$$ direction, we can write the integral as:

$$W = \int_{0}^{4} (2+3x) dx$$

Now we can integrate the function with respect to $$x$$:

$$W = \int_{0}^{4} (2+3x) dx = \int_{0}^{4} 2 dx + \int_{0}^{4} 3x dx$$

$$W = \left[ 2x \right]_0^4 + \left[ \frac{3}{2}x^2 \right]_0^4$$

Now we can plug in the limits of integration:

$$W = (2 \cdot 4 - 2 \cdot 0) + \left(\frac{3}{2} \cdot 4^2 - \frac{3}{2} \cdot 0^2 \right)$$

$$W = 8 + 24$$

$$W = 32 \mathrm{~J}$$

So the work done by the force during the displacement from $$x = 0$$ to $$x = 4 \mathrm{~m}$$ is 32 Joules.