The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions.  What is the area of the bounded region?
The equation given can be rewritten as:

$40|x| = - y^2 - 2xy + 400$
We can split the equation into a piecewise equation by breaking up the [absolute value](https://artofproblemsolving.com/wiki/index.php/Absolute_value):

$40x = -y^2 - 2xy + 400\quad\quad x\ge 0$
$40x = y^2 + 2xy - 400 \quad \quad x < 0$
Factoring the first one: (alternatively, it is also possible to [complete the square](https://artofproblemsolving.com/wiki/index.php/Completing_the_square))

$40x + 2xy  = -y^2 + 400$
$2x(20 + y)=  (20 - y)(20 + y)$
[AIME 1998-3.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1998-3.png) [](https://artofproblemsolving.com/wiki/index.php/File:AIME_1998-3.png)
Hence, either $y = -20$, or $2x = 20 - y \Longrightarrow y = -2x + 20$.
Similarily, for the second one, we get $y = 20$ or $y = -2x - 20$. If we graph these four equations, we see that we get a parallelogram with base 20 and height 40. Hence the answer is $\boxed{800}$.