The problem describes a rectangle $PQRS$ with a square of side $x$ cut out from it. The dimensions of the rectangle are $20$ cm in height and $10 + 10 = 20$ cm in width. The area of the shaded portion (rectangle minus the square) is given as $484 \text{ cm}^2$. We need to find the value of $x$.

1. Problem Description

The problem describes a rectangle $PQRS$ with a square of side $x$ cut out from it. The dimensions of the rectangle are $20$ cm in height and $10 + 10 = 20$ cm in width. The area of the shaded portion (rectangle minus the square) is given as $484 \text{ cm}^2$. We need to find the value of $x$.

2. Solution Steps

The area of the rectangle $PQRS$ is given by:

$\text{Area of rectangle} = \text{length} \times \text{width}$

$\text{Area of rectangle} = 20 \text{ cm} \times 20 \text{ cm} = 400 \text{ cm}^2$.

The area of the square is given by:

$\text{Area of square} = x^2$.

The area of the shaded portion is the area of the rectangle minus the area of the square:

$\text{Area of shaded portion} = \text{Area of rectangle} - \text{Area of square}$

$484 = 400 - x^2$

Now, we solve for $x^2$:

$x^2 = 400 - 484$ is impossible, because $x^2$ cannot be negative. The problem statement is incorrect as stated.

Let's assume the width of rectangle $PQRS$ is $(10 + x + 10) = (20 + x)$ cm and height is 20 cm.

Then the area of rectangle $PQRS$ is:

$A_{\text{rectangle}} = 20(20+x)$

The area of the square is:

$A_{\text{square}} = x^2$

The area of the shaded region is:

$A_{\text{shaded}} = A_{\text{rectangle}} - A_{\text{square}}$

$484 = 20(20+x) - x^2$

$484 = 400 + 20x - x^2$

$x^2 - 20x + 84 = 0$

We can factor this quadratic equation:

$(x - 6)(x - 14) = 0$

So, $x = 6$ or $x = 14$.

3. Final Answer

The possible values of $x$ are $6$ and $14$.

If $x = 6$, the rectangle has dimensions 20 cm by 26 cm and area = 520 $cm^2$. The area of the square is $6^2=36 cm^2$. The shaded area is then $520 - 36 = 484 cm^2$

If $x=14$, the rectangle has dimensions 20 cm by 34 cm and area = 680 $cm^2$. The area of the square is $14^2=196 cm^2$. The shaded area is then $680 - 196 = 484 cm^2$

Final Answer: The values of $x$ are 6 and 14.