Final answer:
The work done by the force F(x) = (-5.0x² + 7.0x) N as the particle moves from x = 2.0 m to x = 5.0 m is found by integrating the force function within the given limits, yielding the numerical value of the work.
Step-by-step explanation:
The amount of work done by a variable force as it acts on a particle moving along the x-axis from one position to another can be calculated using the work integral. In the case at hand, the force acting on the particle is given by F(x) = (-5.0x² + 7.0x) N. To find the work done as the particle moves from x = 2.0 m to x = 5.0 m, we integrate the force function with respect to x between these two limits.
The work done by the force is given by the integral of F(x) dx from 2.0 m to 5.0 m:
W = ∫_{2.0}^{5.0} (-5.0x² + 7.0x) dx
After performing this integration, we obtain the total work done on the particle over the specified distance. This integral can be evaluated either analytically or using mathematical software to get the exact numerical value of the work done.