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In this case, the acceleration of the particle is given by the formula \(a_x = +2x\). This means that the acceleration of the particle is directly proportional to its displacement from the origin. As the particle moves away from the origin, its acceleration increases. This kind of motion can be seen in restoring forces, such as in springs.

The acceleration of particle B is given by the formula \(a_x = +2x^2\). In this case, the acceleration of the particle is directly proportional to the square of the displacement from the origin. This means that the particle's acceleration increases at a faster rate as its displacement increases. Such a relation can be seen in certain nonlinear systems, like charged particles inside an electric field.

For particle C, the acceleration is given by the formula \(a_x = -2x^2\). This relation tells us that the acceleration of the particle is inversely proportional to the square of its displacement from the origin. As the particle moves away from the origin, its acceleration decreases and becomes more negative. This phenomenon can be observed in some systems where a central restoring force opposes the displacement, like in the case of celestial bodies experiencing gravitational force.

The acceleration of particle D is given by the formula \(a_x = -2x\). In this case, the acceleration of the particle is inversely proportional to its displacement from the origin. As the particle moves away from the origin, its acceleration decreases and becomes more negative. Such motion is often associated with negative feedback systems, like dampening forces. In summary, we have analyzed the behavior of four particles based on the given acceleration and displacement relations. Particle A experiences an acceleration directly proportional to its displacement, while particles B and C experience accelerations proportional to the square of their respective displacements. Meanwhile, particle D experiences an acceleration inversely proportional to its displacement.