Differentiate the equation to find the tangent:
2x+6ydy/dx=0. At (3,-1) this becomes 6-6dy/dx=0 so dy/dx=1 and y+1=x-3, y=x-4 is the equation of the tangent line. At (-3,-1) this becomes -6-6dy/dx=0 and dy/dx=-1 and y+1=-(x+3), y=-x-4 is the equation of the tangent line.
The tangent lines intercept one another when x-4=-x-4, so x=0 and y=-4. The intersection is at (0,-4).