Find the equations to the normal lines of a curve where $x = 4$. Find the equations to the normal lines for the curve defined by
  
  $$y^2+xy-x^2=5$$
  
  In the points where $x=4$.


I'm not entirely sure how to even tackle this kind of problem, but let's see:

When $x=4$, we would have

$$y^2+4y-16=5$$

And we would have two solutions: $y=3$ and $y=-7$, so we need the equations for the normal lines that pass through $(4,3)$ and $(4,-7)$.

So I'm already confused. How come this curve has two images for $x = 4$?
I don't think I even understand what I am being asked.
You are given an implicitly defined curve $F(x,y)=0$ and found points $(x_0,y_0)$ on this curve. The tangent at these points is defined by implicit differentiation
$$
∂_xF(x_0,y_0)·(x-x_0)+∂_yF(x_0,y_0)·(y-y_0)=0
$$
which also tells you the direction of the normal to the tangent and thus to the curve.
See, there are two tangent lines for the circle at $x=0$.