The force acting on a particle can be determined from the potential energy function by using the negative gradient. In three-dimensional space, the force vector F is related to the potential energy function $ U $ by:

$$ \mathbf{F} = -\nabla U $$

where $ \nabla U $ (the gradient of $ U $) is a vector with components given by the partial derivatives of $ U $ with respect to $ x $, $ y $, and $ z $:

$$ \nabla U = \left( \frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z} \right) $$

Given the potential energy function:

$$ U = 2x^2 + 3y^3 + 2z $$

Calculate the partial derivative of $ U $ with respect to $ x $ to find the $ x $-component of the force:

Partial Derivative with respect to $ x $:

$$ \frac{\partial U}{\partial x} = \frac{\partial}{\partial x}(2x^2 + 3y^3 + 2z) = 4x $$

Calculate the $ x $-component of the force:

Since $ F_x = -\frac{\partial U}{\partial x} $, the $ x $-component of the force is:

$$ F_x = -4x $$

Evaluate at point $ P(1,2,3) $:

Substitute $ x = 1 $ into the expression for $ F_x $:

$$ F_x = -4(1) = -4 $$

The magnitude of the $ x $-component of the force is:

$$ |F_x| = 4 \, \text{N} $$

Thus, the correct answer is Option A: 4.