How many points of intersection are there between the graphs of the hyperbola and ellipse?

Equate them.....because when they are equal is when they have an intersection/common point

3x^2+4y^2+3x−2y+3=3x^2−2y^2+3x−y+3      delete 3x^2   and 3x from both sides

4y^2 -2y+3 = -2y^2-y+3      gather like terms on one side

6y^2 -y = 0

y(6y-1) = 0      y = 0    or 1/6      now that you have values for 'y'   sub back into one of the original equations to fing the corresponding 'x' values.

SO there is 2 points of intersection.

This is an interesting problem because the first equation:  3x² + 4y² + 3x - 2y + 3 =  0  does not have a real-valued graph.

If you attempt to place this graphing form:      3x² + 3x  + 4y² - 2y + 3 =  0 

                                                                        3(x² + x)  +  4(y² - ½y)  =   -3

Completing the squares:                                 3(x² + x + ¼)  +  4(y² - ½y + 1/16)  =  -3 + 3/4 + 1/4

                                                                        3(x + ½)² + 4(y -  ¼)²  =  -2

This is not a real-valued function.

So, the points of intersection do not exist in a real-valued plane.