Integration over a region/(double integral, how to factor it) Urgent please

[seto6]

Homework Statement

Use a double integral to find the area of the region D, where

D={(x,y) contained in R² | 4x²+12xy+13y²+40y<=-75}

Hint: complete the square
I have a hard time getting the region, i should complete the square to get a formula in R2 that describes the region, i tried to do complete the square but it comes something like this:
(2x+3y)² + 4(y+5)²=25

this does not make sense it look weird, i am not sure how to get this area. please help

The Attempt at a Solution

above

 

[lanedance]

having a look at the form may help understanding, its a rotated ellipse
http://www.wolframalpha.com/input/?i=4*x^2+12*x*y+13*y^2+40*y+=+-75

 

[seto6]

yes i have tried that but, I am not suppose to use any calculators. just by hand there fore some how i must factor not sure how