The ellipse equation you derived is incorrect. Sorry.
It should be:
\(\displaystyle \L\\\frac{(x-3)^{2}}{4}+\frac{(y+1)^{2}}{2}=1\)
It's center is at (3,-1)
Since x has the largest denominator, the major axis lies along the x-axis.
\(\displaystyle a=2, \;\ b=\sqrt{2}\)
The foci lie c units on either side of the center.
You have a and b. Use Pythagoras to find c.