Solution:     

Let the breadth of mango grove be l.

Length of mango grove will be 2l.

Area of mango grove $=\text{ }\left( 2l \right)\text{ }\left( l \right)=\text{ }2{{l}^{2}}$

$2{{l}^{2~}}=\text{ }800$ l² 

$=\text{ }800/2\text{ }=\text{ }400$

${{l}^{2~}}\text{ }400\text{ }=0$

Comparing the given equation with $a{{x}^{2}}~+~bx~+~c~=\text{ }0$ , we get

$a~=\text{ }1,~b~=\text{ }0,~c~=\text{ }400$

As we know, Discriminant $=~{{b}^{2}}~\text{ }4ac$

$=>\text{ }{{\left( 0 \right)}^{2}}~\text{ }4\text{ }\times \text{ }\left( 1 \right)\text{ }\times \text{ }\left( \text{ }\text{ }400 \right)\text{ }=\text{ }1600$

$Here,~{{b}^{2}}~\text{ }4ac~>\text{ }0$

Thus, the equation will have real roots. And hence, the desired rectangular mango grove can be designed.

$l~=\text{ }\pm 20$

As we know, the value of length cannot be negative.

Therefore, breadth of mango grove = 20 m

Length of mango grove = 2 × 20 = 40 m