Start by completing the square.
x^2-8x-3y^2+12y=-16
x^2-8x+16-3(y^2-4y+4)=-16+16-12=-12
Notice the addition of those constants allows us to factor these expressions and because we did it to both sides it’s completely legit.
(x-4)^2-3(y-2)^2=-12
However, it seems we get a negative radius, which is not allowed. I think you entered the problem incorrectly.
