There are two sets of rules and you can go by either one.  

The simpler one is for second degree polynomial equations which have no
xy term. It is a look-and-see rule that requires no calculation:

1. If there is an x² term but no y² term or a y² term but no x² term,
   the graph of the  equation is a parabola.

2. If the x² term and the y² term have equal coordinates when on the same
   side of the equal sign, the graph of the equation is a circle.

3. If the x² term and the y² term have the same sign when on the same
   side of the equal sign, the graph of the  equation is an ellipse.
   (Note than a circle is a special case of an ellipse when the x² and y²
    terms not only have the same sign, but also the same coefficient).

4. If the x² term and the y² term have opposite signs when on the same
   side of the equal sign, the graph of the equation is a hyperbola.
     
Your equation has no xy term and fits 3, so it is an ellipse.

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There is a more general rule that applies to all second degree polynomials
regardless of whether they have an xy term or not.

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The more general rule requires a calculation:

Put the equation in the form

Ax² + Bxy + Cy² + Dx + Ey + F = 0

then form the discriminant B²-4AC

If this is 0, the graph of the equation is a parabola.
If this is negative , the graph of the equation is an ellipse (or circle).
If this is positive , the graph of the equation is a hyperbola.


 

1x² + 0xy + y² + 6x + 0y + 4 = 0

A=1  B=0  C=  D=6  E=0  F=4

B²-4AC = 0²-4(1)(} = 

That is negative so the equation is of an ellipse by the general rule.

Edwin