To find: The dimension of the froor and the dimension of the fountain-pool area.

Procedure used:
And Application of a Nonlinear System:
"Step 1: Assume the variables in order to represent unknown quantities.
Step 2: Write a system of equations modeling the condition in the problem.
Step 3: Solve the system and answer the question asked in the problem.
Step 4: Check the proposed solution in the original wording of the problem."
Calculation:
It is given that the perimeter and area of the floor with the fountain-pool area removed are 24 meters and 21 square meter, respectively.
Step 1: Assume the side of floor and side fountain-pool floor to be x and y, respectively.
Step 2: Model the given conditions into a system of equations.
As the perimeter of the floor without fountain-pool area is 24 meters, it gives ${\displaystyle 4x+2y=24(3x+x-y+3y=4x+2y)}$
As the area of the floor without fountain-pool area is 21 square meter, it gives ${\displaystyle x^2-y^2=21}$.
Thus, the system of equations becomes $\{\begin{array}{l}x+y=20\\ xy=96\end{array}$
Step 3: Modify the equation ${\displaystyle 4x+2y=24}$ as ${\displaystyle y=\frac{24-4x}{2}}$
Substitute ${\displaystyle y=\frac{24-4x}{2}}$ in the equation ${\displaystyle x^2-y^2=21}$ and obtain the quadratic equation as,
${\displaystyle x^2-{\left(\frac{24-4x}{2}\right)}^2=21}$
${\displaystyle x^2-\frac{576-192x+16x^2}{4}=21}$
${\displaystyle 4x^2-576+192x-16x^2=84}$
${\displaystyle -12x^2+192x-660=0}$
That further becomes ${\displaystyle x^2-16x+55=0}$.
Solve the equation ${\displaystyle x^2-16x+55=0}$ and obtain the value of x as follows.
${\displaystyle x^2-16x+55=0}$
${\displaystyle x^2-11x-5x+55=0}$
${\displaystyle x(x-11)-5(x-11)=0}$
${\displaystyle (x-11)(x-5)=0}$
On further simplification,
${\displaystyle x-11=0}$ or ${\displaystyle x-5=0}$
${\displaystyle x=11}$ or ${\displaystyle x=5}$
Substitute ${\displaystyle x=11}$ in the equation ${\displaystyle 4x+2y=24}$ and obtain the value of y as follows.
${\displaystyle 4x+2y=24}$
${\displaystyle 4\left(11\right)+2y=24}$
${\displaystyle 44+2y=24}$
${\displaystyle 2y=-20}$
${\displaystyle y=-10}$
Substitute ${\displaystyle x=5}$ in the equation ${\displaystyle 4x+2y=24}$ and obtain the value of y as follows.
${\displaystyle 4x+2y=24}$
${\displaystyle 4\left(5\right)+2y=24}$
${\displaystyle 20+2y=24}$
${\displaystyle 2y=4}$
${\displaystyle y=2}$
Thus, for ${\displaystyle x=11,y=-10}$, and for ${\displaystyle x=5,y=2}$.
The length cannot be negative. Therefore, the solution is (5,2)
Step 4: Check the result, by substituting the obtained solutions in the given original equations