Question 19417
You will need to get your equation into the standard form for an ellipse: 
{{{((x-h)^2)/a + ((y-k)^2)/b = 1}}}
You can do this by completing the square in the x-terms and the y-terms. Start by groupng the x-terms together and grouping the y-terms together.

{{{(9x^2 - 18x) + (4y^2 + 8y) = 23}}} Now,factor a 9 from the x-group and factor a 4 from the y-group.
{{{9(x^2 - 2x) + 4(y^2 + 2y) = 23}}} Complete the square in the x-terms within the parentheses by adding the square of half the x-coefficient to both sides of the equation, and do the same for the y-terms.
{{{9(x^2 - 2x + 1) + 4(y^2 + 2y + 1) = 23 + 9 + 4}}}
{{{9(x^2 - 2x + 1) + 4(y^2 + 2y + 1) = 36}}} Now divide both sides by 36.
{{{(x^2 - 2x + 1)/4 + (y^2 + 2y + 1)/9 = 1}}} Factor the denominators.
{{{((x - 1)^2)/4 + ((y + 1)^2)/9 = 1}}}
This is the standard form of the equation for an ellipse with the center at (1, -1) and major axis is parallel to the y-axis. The length of the major axis is 3 and the length of the minor axis is 2