if a moving point p is always equidistant from the point A(2,4) and the line y=-4, find the equation of the locus of p

By definition, that would be a parabola.

Using that definition, let any point on the locus be P(x,y)
then
√( (x-2)^2 + (y-4)^2 ) = √( (x-x)^2 + (y+4)^2)
square both sides and expand
x^2 - 4x + 4 + y^2 - 8y + 16 = y^2 + 8y + 16
x^2 - 4x + 4 = 16y

looks like the equation of a standard upwards opening parabola to me
Doctor it up any way you have to