The latus rectum is a chord joining two points on the parabola which are at a distance 2a from the focus. It passes through the focus and is parallel to the directrix.

The distance between the two points (-4, 1) and ( 2,1) is 6. This is equal to 4*a.

4a = 6

=> a = 6/4 = 3/2.

The vertex is arrived at as ((-4 + 2)/2, (1 - 3/2)) = (-1 , -1/2).

The equation of the parabola is (x + 1)^2 = 4*(3/2)*(y + 1/2)

=> x^2 + 1 + 2x = 6y +3

=> x^2 + 2x - 6y - 2 = 0

The required equation of the parabola is x^2 + 2x - 6y - 2 = 0.