Interpret the given position and velocity

The object moves in a horizontal circle centered at the origin.

When it is at $x=+2\,\text{m}$, on the circular path this means the point is $(2,0).$

Its velocity there is given as $\vec v = -4\hat j\ \text{m/s}.$

So at the rightmost point of the circle, the velocity is downward.

Find the radius and speed

Since the center is at the origin and the point is $(2,0)$, the radius is $r = 2\,\text{m}.$

Speed is constant and equal to the magnitude of velocity: $v = 4\,\text{m/s}.$

Determine the direction of motion

At $(2,0)$, the velocity is downward. That means the particle is moving clockwise.

Find the acceleration at $(2,0)$ to confirm circular motion

In uniform circular motion, acceleration is centripetal, i.e. toward the center.

At $(2,0)$, toward the origin is in the $-\hat i$ direction.

Magnitude of centripetal acceleration: $a = \frac{v^2}{r} = \frac{4^2}{2} = 8\,\text{m/s}^2.$

So there, $\vec a = -8\hat i\ \text{m/s}^2.$

Now find velocity and acceleration at $x=-2\,\text{m}$

The point $x=-2\,\text{m}$ on the circle is $(-2,0).$

For clockwise motion, at the leftmost point the velocity is upward, so $\vec v = +4\hat j\ \text{m/s}.$

Find acceleration at $(-2,0)$

Centripetal acceleration always points toward the center.

From $(-2,0)$ toward the origin is in the $+\hat i$ direction.

Magnitude remains $8\,\text{m/s}^2.$

Therefore, $\vec a = +8\hat i\ \text{m/s}^2.$

Match with the options

This corresponds to: $\vec v = 4\hat j\ \text{m/s},\quad \vec a = 8\hat i\ \text{m/s}^2.$

So the correct option is B.