Identify the conic section $$$5 \left(x - 3\right)^{2} = 20 - 4 \left(y - 6\right)^{2}$$$

The general equation of a conic section is $$$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$$$.

In our case, $$$A = 5$$$, $$$B = 0$$$, $$$C = 4$$$, $$$D = -30$$$, $$$E = -48$$$, $$$F = 169$$$.

The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = -1600$$$.

Next, $$$B^{2} - 4 A C = -80$$$.

Since $$$B^{2} - 4 A C \lt 0$$$, the equation represents an ellipse.

To find its properties, use the ellipse calculator.

$$$5 \left(x - 3\right)^{2} = 20 - 4 \left(y - 6\right)^{2}$$$A represents an ellipse.

General form: $$$5 x^{2} - 30 x + 4 y^{2} - 48 y + 169 = 0$$$A.

Graph: see the graphing calculator.