The problem states that a rectangle $PQRS$ has a square of side $x$ cut out from two of its corners. The dimensions of the rectangle are $20$ cm by $10 + 10 = 20$ cm. The area of the shaded portion is $484$ cm$^2$. We need to find the value(s) of $x$.

1. Problem Description

The problem states that a rectangle $PQRS$ has a square of side $x$ cut out from two of its corners. The dimensions of the rectangle are $20$ cm by $10 + 10 = 20$ cm. The area of the shaded portion is $484$ cm$^2$. We need to find the value(s) of $x$.

2. Solution Steps

First, we find the area of the entire rectangle $PQRS$.

The length of $PQRS$ is 20 cm and the width is 20 cm, so the area is $20 \times 20 = 400$ cm$^2$.

Two squares with sides $x$ cm are removed from the rectangle. The area of each square is $x^2$ cm$^2$, so the combined area of the two squares is $2x^2$ cm$^2$.

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

We are given that the shaded area is $484$ cm$^2$. However, this number appears to be greater than the total area of the rectangle. I'll assume it is a typo and work with the idea that the area of the shaded region is less than the total rectangle area.

Let's suppose the area of the shaded portion equals the area of rectangle $PQRS$ MINUS the areas of the two squares of side length $x$.

The area of rectangle $PQRS$ is $20 \times 20 = 400$.

The area of the two squares is $2x^2$.

So, the shaded area is $400 - 2x^2$.

The problem states that the shaded area is 484 cm$^2$.

Therefore, $400 - 2x^2 = 484$.

Rearranging the equation gives $2x^2 = 400 - 484$.

So, $2x^2 = -84$.

$x^2 = -42$. Since $x$ represents a length, it must be a real number. This equation has no real solutions because $x^2$ cannot be negative. Therefore, there is an error in the problem statement. We are likely supposed to have area of the shaded region be a smaller number.

Let us suppose the area of the shaded region is $316 cm^2$. Then:

$400 - 2x^2 = 316$.

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

$x^2 = 42$.

$x = \sqrt{42}$.

3. Final Answer

Assuming the area of the shaded region is 316 cm$^2$ (instead of 484 cm$^2$), the value of $x$ is $\sqrt{42}$ cm.

Since the stated area of the shaded region is 484 cm$^2$, and this leads to no real solution for $x$, we can say that there are no real values of $x$ that satisfy the problem. However, there could have been a typo, so let's just present our equation.

Final Answer: $400 - 2x^2 = 484$, which implies $x^2 = -42$. No real solutions for x.