The area enclosed by the graph of ${\mathrm{y}}^2+2\mathrm{xy}+40\left|\mathrm{x}\right|=400$ (in square units ) is

The given curve is ${\mathrm{y}}^2+2\mathrm{xy}+40\left|\mathrm{x}\right|=400$

The given equations can be written as $y^2+2xy+40x=400$ and $y^2+2xy-40x=400$

$\begin{array}{rcl}{y}^2+2xy+40x-400& =& 0\\ y^2-400+2xy+40x& =& 0\\ \left(y-20\right)\left(y+20\right)+2x\left(y+20\right)& =& 0\\ \left(y+20\right)\left(y-20+2x\right)& =& 0\\ \left(y+20\right)\left(2x+y-20\right)& =& 0\end{array}$

Similarly 

$\begin{array}{rcl}{y}^2+2xy-40x-400& =& 0\\ y^2-400+2xy-40x& =& 0\\ \left(y-20\right)\left(y+20\right)+2x\left(y-20\right)& =& 0\\ \left(y-20\right)\left(y+20+2x\right)& =& 0\\ \left(y-20\right)\left(2x+y+20\right)& =& 0\end{array}$

The given equation represents four lines $y-20=0,y+20=0,2x+y-20=0$ and $2x+y+20=0$

The rough sketch of the area of the required region is as below

Therefore, the required area is area of the parallelogram of base 20 units and height 40 units as shown in the figure

therefore, the area of the parallelogram is $20\times 40=\boxed{800 \text{sq units}}$