The diagram shows a rectangle $PQRS$ with length $10+10=20$ cm and width $20$ cm. A square of side $x$ cm has been cut out twice from the rectangle. The area of the shaded portion is $484$ $cm^2$. We need to find the value of $x$.

1. Problem Description

The diagram shows a rectangle $PQRS$ with length $10+10=20$ cm and width $20$ cm. A square of side $x$ cm has been cut out twice from the rectangle. The area of the shaded portion is $484$ $cm^2$. We need to find the value of $x$.

2. Solution Steps

The area of the rectangle $PQRS$ is given by

$Area_{rectangle} = length \times width = 20 \times 20 = 400 \ cm^2$

The area of each square is $x^2$. Since there are two squares cut out, the total area of the squares is $2x^2$.

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

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

$484 = 400 + 2 \times 20 \times x - 2x^2 = 400 - 2x^2$. This seems incorrect.

However, the given diagram shows $Area_{shaded} = Area_{rectangle} - 2x^2$.

$Area_{shaded} = 484$

Therefore,

$400 - 2x^2 = 484$

$-2x^2 = 484 - 400$

$-2x^2 = 84$

$x^2 = -42$

This is not possible, since $x$ is a length and $x^2$ must be positive.

The problem is that the diagram is misleading. The diagram does NOT have the square being "cut from" the rectangle. The problem states, the diagram shows a rectangle PQRS from which a square of side x cm has been cut.

Let's compute the area of the rectangle to be $20 \times (10+10) = 400$.

The area of the shaded portion is $484$.

The formula we need is the Area of the Rectangle - 2 times the area of the square = Area of the Shaded Portion.

$400-2x^2 = 484$

$2x^2 = 400 - 484$

$2x^2 = -84$

$x^2 = -42$.

This is incorrect.

The area of the rectangle is $20 \times 20 = 400 cm^2$. The squares of side $x$ are removed from the rectangle. So,

$400 - 2x^2 = 484$

$2x^2 = 400 - 484 = -84$

$x^2 = -42$

This is not possible. There must be an error in the problem description.

Assume that the area of the shaded region is $316 \ cm^2$ instead of $484 \ cm^2$.

$400 - 2x^2 = 316$

$2x^2 = 400 - 316 = 84$

$x^2 = 42$

$x = \sqrt{42}$

Let us re-evaluate the prompt. If the area of the shaded portion is 316 $cm^2$, find the value of x. Then

Area of rectangle PQRS - 2*Area of square = Area of shaded region.

$400 - 2x^2 = 316$

$2x^2 = 400 - 316$

$2x^2 = 84$

$x^2 = 42$

$x = \sqrt{42} \approx 6.48$

However, with area of $484$:

$400-2x^2 = 484$

$2x^2 = -84$

So, the original problem is wrong.

3. Final Answer

There is likely an error in the problem description, as the calculation leads to a negative value for $x^2$.

If the area of the shaded portion were $316$ $cm^2$, then $x = \sqrt{42}$.

Final Answer: No real solution.