To find the foci of an ellipse, we need to find the value of c in the standard form equation of an ellipse:
(x-h)^(2)/(a^(2)) + (y-k)^(2)/(b^(2)) = 1
where (h, k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Comparing the given equation:
((x-2)^(2))/(144) + ((y+3)^(2))/(225) = 1
to the standard form equation, we can see that the center is (2, -3), a = 12 (since 12^2 = 144), and b = 15 (since 15^2 = 225).
To find c, we can use the equation:
c = sqrt(a^(2) - b^(2))
c = sqrt(12^(2) - 15^(2))
c = sqrt(144 - 225)
c = sqrt(-81)
Since c = sqrt(-81), which is an imaginary number, it means that the ellipse does not have real foci.