ANSWER
D. Ellipse;
[tex]3{x}^{2} +{y}^{2} + 6x - 6y + 3= 0[/tex]
EXPLANATION
The given equation is
[tex]3 {x}^{2} + {y}^{2} = 9[/tex]
Dividing through by 9 gives
[tex] \frac{ {x}^{2} }{ 3} + \frac{ {y}^{2} }{9} = 1[/tex]
This is the equation of an ellipse centered at the origin.
If this ellipse has been translated, so that its center is now at (-1,3), then the equation of the translated ellipse becomes
[tex]\frac{ {(x + 1) }^{2} }{ 3} + \frac{ {(y - 3)}^{2} }{9} = 1[/tex]
We multiply through by 9 to get,
[tex]3 {(x + 1)}^{2} + {(y - 3)}^{2} = 9[/tex]
Expand to obtain;
[tex]3( {x}^{2} + 2x + 1) + {y}^{2} - 6y + 9 = 9[/tex]
Expand to obtain;
[tex]3{x}^{2} + 6x + 3+ {y}^{2} - 6y + 9 = 9[/tex]
Regroup and equate to zero to obtain;
[tex]3{x}^{2} +{y}^{2} + 6x - 6y + 3= 0[/tex]
