Horizontal Tangent - Implicit Differentiation

a curve in the plane is given by the following equation

x^4+2x^2y+3y^2=6

A) Find an expression for dy/dx using implicit differentiation

B) Find the four points on the curve where the tangent line is horizontal

Ok I know how to solve for the derivative. I got

dy/dx = (4x^3-4xy)/(2x^2+6y)

However Im not quite sure how to solve when the tangent is horizontal. I that the tangent is horizontal when the derivative equals 0. However I don't know how solve for this when it is an implicit derivative.