The positions of two particles, A and B, moving along a straight...

The position of particle A is given by $x_A = 4t^2 + 2t - 5$

The position of particle B is given by $x_B = 4t^2 + 2t + 3$

The relative position of particle A with respect to particle B is:

$x_{AB} = x_A - x_B$

$x_{AB} = (4t^2 + 2t - 5) - (4t^2 + 2t + 3) = -8 \text{ m}$

The relative velocity of particle A with respect to particle B is:

$v_{AB} = \dfrac{dx_{AB}}{dt} = \dfrac{d}{dt}(-8) = 0 \text{ m/s}$

Since the relative velocity is zero at all times, particle A is permanently at rest with respect to particle B.

Answer: Particle A is permanently at rest with respect to particle B.