Solutions:

(i) Let us consider,

Breadth of the rectangular plot = x m

Thus, the length of the plot = (2x + 1) m.

As we know,

$Area\text{ }of\text{ }rectangle\text{ }=\text{ }length~\times ~breadth\text{ }=\text{ }528\text{ }{{m}^{2}}$

Putting the value of length and breadth of the plot in the formula, we get,

$\left( 2x~+\text{ }1 \right)\text{ }\times ~x~=\text{ }528$

$\Rightarrow 2{{x}^{2}}~+~x~=528$

$\Rightarrow 2{{x}^{2}}~+~x-528\text{ }=\text{ }0$

Therefore, the length and breadth of plot, satisfies the quadratic equation, $2{{x}^{2}}~+~x-528\text{ }=\text{ }0$, which is the required representation of the problem mathematically.

(ii) Let us consider,

The first integer number = x

Thus, the next consecutive positive integer will be = x + 1

Product of two consecutive integers $=~x~\times ~\left( x~+1 \right)\text{ }=\text{ }306$

$\Rightarrow ~{{x}^{2~}}+~x~=\text{ }306$

$\Rightarrow ~{{x}^{2~}}+~x-306\text{ }=\text{ }0$

Therefore, the two integers x and x+1, satisfies the quadratic equation, ${{x}^{2~}}+~x-306\text{ }=\text{ }0$ , which is the required representation of the problem mathematically.