A rectangle $PQRS$ has dimensions 20 cm by 10 cm. A square of side $x$ cm is cut from each of two corners of the rectangle. The area of the remaining shaded region is 484 $cm^2$. We need to find the value of $x$.

1. Problem Description

A rectangle $PQRS$ has dimensions 20 cm by 10 cm. A square of side $x$ cm is cut from each of two corners of the rectangle. The area of the remaining shaded region is 484 $cm^2$. We need to find the value of $x$.

2. Solution Steps

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

$Area_{rectangle} = length \times width$

$Area_{rectangle} = 20 \times 10 = 200$ $cm^2$

The area of each square that is cut out is:

$Area_{square} = side \times side$

$Area_{square} = x^2$ $cm^2$

Since there are two squares, the total area of the two squares is $2x^2$ $cm^2$.

The area of the shaded region is the area of the rectangle minus the area of the two squares.

$Area_{shaded} = Area_{rectangle} - 2 \times Area_{square}$

We are given that the area of the shaded region is 484 $cm^2$. Therefore,

$484 = 200 - 2x^2$

This equation is incorrect.

The problem states a rectangle $PQRS$ from which a square of side $x$ has been cut.

The diagram shows TWO squares cut from the corners. So the correct equation is:

$Area_{shaded} = Area_{rectangle} - 2 Area_{square}$

$484 = 20 \times 10 - 2x^2$

$484 = 200 - 2x^2$

$2x^2 = 200 - 484$

$2x^2 = -284$

This implies that $x^2$ would be negative, which is not possible.

Let's examine the problem description and the diagram again.

Rectangle $PQRS$ has length 20 cm and width 10 cm, area = 200 $cm^2$.

Two squares of side $x$ cm are cut. Thus the area of the two squares is $2x^2$.

The area of the shaded region is given as 184 $cm^2$. Thus:

$200 - 2x^2 = 184$

$2x^2 = 200 - 184 = 16$

$x^2 = 8$

$x = \sqrt{8} = 2\sqrt{2}$

The original text states "The area of the shaded portion is 484 $cm^2$". The diagram looks as if the shaded region is slightly smaller than the unshaded. Therefore, the number 484 may be a misprint. However, the problem text is important, so we must go with it. Let's try again.

$200 - 2x^2 = 484$ is not correct. The error is that the squares have been added instead of subtracted.

Looking at the diagram, it looks as if two rectangles are subtracted. If 10 cm and 10 cm are deducted from 20 cm and the shaded area is 184 $cm^2$. This seems plausible.

Let's assume that the correct area of the shaded portion is

1

8

4. $200 - 2x^2 = 184$

$2x^2 = 16$

$x^2 = 8$

$x = \sqrt{8} = 2\sqrt{2}$

However, since we are asked to solve the problem as it is, the calculation with 484 is important:

$200 - 2x^2 = 484$

$-2x^2 = 284$

$2x^2 = -284$

Since a negative value does not make sense, there might be a typo in the question, but to answer the question with the information as is.

Area of rectangle - 2 * area of squares = area of shaded region

20 * 10 - 2 * x * x = 484

200 - 2x^2 = 484

-2x^2 = 284

x^2 = -142

x = sqrt(-142)

x does not exist, since there is no real number $x$ such that $x^2 = -142$.

Since $2x^2$ cannot be greater than 200, it must be a misprint in the problem

If we assume that the total area of shaded region is 184:

$200 - 2x^2 = 184$

$2x^2 = 16$

$x^2 = 8$

$x = \sqrt{8} = 2 \sqrt{2} \approx 2.83$

3. Final Answer

There is no real solution for $x$ given the problem. If the area of the shaded region was 184 $cm^2$ instead of 484 $cm^2$, then $x = 2\sqrt{2}$ cm. Since the problem says the area is 484, the area of the squares would be negative. There seems to be an error in the question, therefore, it is not solvable.

Since we are asked to find the value of $x$, and given the question, we can conclude that the problem has no solution.

$x = \sqrt{-142}$ (This is not a real solution, but we follow the directions given). Since the value of $x$ is non-real, we cannot provide a final answer for this problem.