Given force

The force acts along the $x$-direction: $\vec F=(2+3x)\hat i$

So the magnitude of force as a function of position is $F(x)=2+3x$

Formula for work done by a variable force

Work done from $x=0$ to $x=4$ is $W=\int_{0}^{4} \vec F\cdot d\vec r$

Since both force and displacement are along the $x$-axis, $\vec F\cdot d\vec r = F(x)\,dx$

Therefore, $W=\int_0^4 (2+3x)\,dx$

Integrate

$\int (2+3x)\,dx = 2x+\frac{3x^2}{2}$

Now apply the limits $0$ to $4$: $W=\left[2x+\frac{3x^2}{2}\right]_0^4$

At $x=4$: $2(4)+\frac{3(4)^2}{2}=8+\frac{3\cdot 16}{2}=8+24=32$

At $x=0$: $2(0)+\frac{3(0)^2}{2}=0$

Hence, $W=32-0=32\text{ J}$

Final answer

The work done is $\boxed{32}$