Identify the given force function and parameters.

The force acting on the object is given by $F(x) = ax + bx^2$.
The given values are $a = 5 \, \text{N/m}$ and $b = -2 \, \text{N/m}^2$.
So, the force function is $F(x) = 5x - 2x^2$.
The object moves from $x_1 = 1 \, \text{m}$ to $x_2 = 3 \, \text{m}$.

The work done ($W$) by a variable force $F(x)$ as an object moves from $x_1$ to $x_2$ is given by the definite integral:

First, find the antiderivative of $F(x)$:

Now, evaluate the definite integral:

To subtract, find a common denominator:

The work done by the force is $\boxed{\frac{8}{3} \text{ J}}$.

The force function is $F(x) = 5x - 2x^2 = x(5 - 2x)$.
The force is zero when $x=0$ or $5-2x=0 \Rightarrow x=2.5$.
Since $x=2.5$ is within the interval $[1, 3]$:

• For $x < 2.5$ (e.g., $x=1$), $F(1) = 5(1) - 2(1)^2 = 3 \, \text{N}$ (positive).
• For $x > 2.5$ (e.g., $x=3$), $F(3) = 5(3) - 2(3)^2 = 15 - 18 = -3 \, \text{N}$ (negative). Thus, the force $F(x)$ does change sign within the interval $[1, 3]$.

Yes, the process of definite integration inherently accounts for the fact that the force $F(x)$ changes sign in the interval. When $F(x)$ is positive, it contributes positive work (force acts in the direction of displacement). When $F(x)$ is negative, it contributes negative work (force acts opposite to the direction of displacement). The definite integral sums these positive and negative contributions algebraically to yield the net work done over the entire interval.