The graph of $y^2+2xy+40|x|=400$ partitions the plane into several regions.  What is the area of the bounded region?
1. **Case 1: \( x \geq 0 \)**
   - When \( x \geq 0 \), \( |x| = x \). The equation becomes:
     \[
     y^2 + 2xy + 40x = 400
     \]
   - Rearrange the equation:
     \[
     y^2 + 2xy + 40x - 400 = 0
     \]
   - Factor the equation:
     \[
     (y + 20)(y - 20) = 2x(y + 20)
     \]
   - This gives two cases:
     - \( y + 20 = 0 \Rightarrow y = -20 \)
     - \( 2x = 20 - y \Rightarrow y = -2x + 20 \)

2. **Case 2: \( x < 0 \)**
   - When \( x < 0 \), \( |x| = -x \). The equation becomes:
     \[
     y^2 + 2xy - 40x = 400
     \]
   - Rearrange the equation:
     \[
     y^2 + 2xy - 40x - 400 = 0
     \]
   - Factor the equation:
     \[
     (y + 20)(y - 20) = -2x(20 - y)
     \]
   - This gives two cases:
     - \( y = 20 \)
     - \( y = -2x - 20 \)

3. **Graphing the Rays**
   - The four rays are:
     - \( y = -20 \)
     - \( y = -2x + 20 \)
     - \( y = 20 \)
     - \( y = -2x - 20 \)
   - These rays form a parallelogram in the coordinate plane.

4. **Calculating the Area of the Parallelogram**
   - The base of the parallelogram is the distance between \( y = -20 \) and \( y = 20 \), which is \( 40 \).
   - The height of the parallelogram is the distance between the points where the lines \( y = -2x + 20 \) and \( y = -2x - 20 \) intersect the x-axis. Solving for \( y = 0 \):
     \[
     0 = -2x + 20 \Rightarrow x = 10
     \]
     \[
     0 = -2x - 20 \Rightarrow x = -10
     \]
   - The height is \( 10 - (-10) = 20 \).

5. **Area of the Parallelogram**
   - The area \( A \) of the parallelogram is given by:
     \[
     A = \text{base} \times \text{height} = 40 \times 20 = 800
     \]

The final answer is \( \boxed{800} \).