I think you got it all right, except for the sign of the coefficient of the y squared term. This parabola opens to the left and is in the 2nd and 3rd quadrants. So the sign has to be negative. It's -1/14, not +1/14
A parabola is the collection of all points (x,y) equidistant from a point (the focus) and a line (the directrix).
Let (x,y) be a point on the parabola, any point.
The distance from (x,y) to x=3 is the same as the distance from (x,y) to (-4,0)
Using the definition of a parabola, find an expression for the distance from (x,y) to the direction line
and to the focus point and set them equal
the distance formula for two points is d =( (x-x1)2 + (y-y1)2)1/2
substituting for the focus = square root of (x+4)2+y2 set this equal to the distance from (x,y) to the directrix line or square root of (x+4)2 + y2 = 3-x
square both sides to get (x+4)2 + y2 = 9-6x+x2
expand left side and cancel x2 terms
x2 + 8x + 16 + y2 = x2 - 6x + 9
14x=-y2-7 divide by 14 x=-y2/14-1/2 the coefficient of the y2 term is -1/14 not +1/14
Other ways to work this, but you got a sign wrong somewhere