Answer
$\frac{(x-3)^2}{4}+\frac{(y+4)^2}{25}=1$, foci: $(3,-4\pm\sqrt {21})$.
Work Step by Step
Step 1. Rewrite the given equation as
$25(x^2-6x+9)+4(y^2+8y+16)=225+64-189=100$
or
$\frac{(x-3)^2}{4}+\frac{(y+4)^2}{25}=1$,
Step 2. We have $a^2=25, b^2=4$ and $c=\sqrt {a^2-b^2}=\sqrt {21}$. The ellipse is centered at $(3,-4)$ with a vertical major axis.
Step 3. We can graph the equation as shown in the figure with foci at $(3,-4\pm\sqrt {21})$.
