Answer
See below
Work Step by Step
Given $12x^2+45y^2+120x+90y-150=0$
We can see that $a=9\\b=0\\c=-4$
We will find the discriminant of the given equation $=b^2-4ac\\=0^2-4(9)(-4)\\=-2160$
Since the discriminant $-2160\lt0$, the conic is an ellipse.
To graph the ellipse, first complete the square in x.
$12x^2+45y^2+120x+90y-150=0\\12(x^2+10x+25)-300+45(y^2+2y+1)-45+7=150\\12(x+5)^2+45(y+1)^2=495\\\frac{(x+5)^2}{41.25}+\frac{(y+1)^2}{11}=1$
From the equation, you can see that the center is at $(-5,-1)$