Such equation of ellipse is of the type #(x-h)^2/a^2+(y-k)^2/b^2=1#, for hyperbola it is of type #(x-h)^2/a^2-(y-k)^2/b^2=1# and for parabola it is #(y-k)=a(x-h)^2# or #(x-h)=a(y-k)^2#.
As we have coefficients of both #x^2# and #y^2# of same sign, it appears to be an ellipse. Let us convert this into desired form.
#4x^2+9y^2-16x+18y-11=0# can be written as
#4x^2-16x+9y^2+18y-11=0#
or #4(x^2-4x+4)+9(y^2+2y+1)=11+16+9#
or #4(x-2)^2+9(y+1)^2=36#
or #(x-2)^2/9+(y+1)^2/4=1#
or #(x-2)^2/3^2+(y+1)^2/2^2=1#
which is an ellipse with center at #(2,1)#, major axis parallel to #x#-axis is #6# and minor axis parallel to #y#-axis is #4# (double of #a# and #b# respectively).
graph{(4x^2+9y^2-16x+18y-11)((x-2)^2+(y+1)^2-0.01)=0 [-3.29, 6.71, -3.36, 1.64]}
