What is the formula for a circle that has a center at (-1,4) and goes through the point (-4,0)?

 Its (x + 1)^2 + (y - 4)^2  = r^2   ( because  the center is at (-1,4)
To find r^2 we substitute for the point (-4,0):-
r^2 = (-4+1)^2 + (0-4)^2 =  9 + 16 = 25

r^2 = 25 so the answer is:-

(x + 1)^2 + (y - 4)^2 = 25