How do I graph #16x^2+y^2+32x-18y=119# algebraically?

This looks like the equation of a circle. The best way to get these into a graphable form is to play around with the equation and complete squares. Let's first regroup these...
#(16x^2 + 32x)+(y^2-18y)=119#

Now take out the factor of 16 in the x "group".
#16(x^2 + 2x) + (y^2-18y)=119#

Next, complete the squares
#16(x^2+2x+1)+(y^2-18y+81)=119+16+81#
#16(x+1)^2+(y-9)^2=216#

Hmm... this would be the equation of a circle, except there's a factor of 16 in front of the x group. That means it must be an ellipse.
An ellipse with center (h, k) and a horizontal axis "a" and vertical axis "b" (regardless of which one is the major axis) is as follows:

#(x-h)^2/a+(y-k)^2/b = 1#

So, let's get this formula into that form.
#(x+1)^2/13.5 + (y-9)^2/216 = 1# (Divide by 216) That's it!

So, this ellipse is going to be centered at (-1, 9). Also, the horizontal axis will have a length of #sqrt13.5# or about #3.67#, and the vertical axis (also the major axis of this ellipse) will have a length of #sqrt216# (or #6sqrt6#), or about #14.7#.

If you were to graph this by hand, you would draw a dot at (-1, 9), draw a horizontal line extending about 3.67 units on either side of the dot, and a vertical line extending about 4.7 units on either side of the dot. Then, draw an oval connecting the tips of the four lines.

If this doesn't make sense, here's a graph of the ellipse.
graph{16x^2 + y^2+32x-18y =119 [-34.86, 32.84, -8, 25.84]}