1. The problem is to find the equation of the ellipse given its center and axes. 2. The ellipse is centered at $(3,2)$. 3. The semi-major axis length along the x-direction is $5$, so $a=5$. 4. The semi-minor axis length along the y-direction is $3$, so $b=3$. 5. The standard form of an ellipse centered at $(h,k)$ with semi-major axis $a$ along x and semi-minor axis $b$ along y is: $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ 6. Substitute $h=3$, $k=2$, $a=5$, and $b=3$: $$\frac{(x-3)^2}{25} + \frac{(y-2)^2}{9} = 1$$ This is the equation of the ellipse. Final answer: $$\frac{(x-3)^2}{25} + \frac{(y-2)^2}{9} = 1$$