The force acting on the particle is given by F(x) = -kx^3. According to Newton's second law, F = ma, where m is the mass of the particle and a is its acceleration. Acceleration is the derivative of velocity with respect to time, a = dv/dt. Therefore, we can write the equation as:

m * dv/dt = -kx^3

We can rearrange this equation to separate the variables:

dv = -kx^3 * dt / m

Integrating both sides with respect to their variables gives:

v = -k/(4m) * x^4 + C

where C is the constant of integration. This is the expression for the velocity as a function of position.

The total mechanical energy of the particle is given by the sum of its kinetic and potential energy. The kinetic energy (K) is given by (1/2)mv^2 and the potential energy (U) is given by the integral of F(x)dx, which in this case is (-k/4)x^4. Therefore, the total mechanical energy (E) is:

E = K + U = (1/2)mv^2 + (-k/4)x^4

As the particle moves through the potential, its velocity and position will change according to the derived expression. If the particle starts at a position x with a certain amount of mechanical energy, it will speed up as it moves towards the origin (since the force is negative and thus pulling it towards the origin), reach a maximum speed at the origin, and then slow down as it moves away from the origin (since the force becomes positive and thus pushes it away from the origin). The total mechanical energy will remain constant throughout this motion, as energy is conserved.