Questions: Find the area of the shaded region. The area of the shaded region is (Simplify your answer.)

Find the area of the shaded region. The area of the shaded region is (Simplify your answer.)
Transcript text: Find the area of the shaded region. The area of the shaded region is $\square$ (Simplify your answer.) $\square$
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Solution

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Solution Steps

Step 1: Understand the Problem

We need to find the area of the shaded region. The shaded region is the larger square minus the smaller square.

Step 2: Calculate the Area of the Larger Square

The side length of the larger square is given as \( (7x - 4) \) inches. The area of a square is side length squared. \[ \text{Area of larger square} = (7x - 4)^2 \]

Step 3: Calculate the Area of the Smaller Square

The side length of the smaller square is given as \( x \) inches. The area of a square is side length squared. \[ \text{Area of smaller square} = x^2 \]

Step 4: Subtract the Area of the Smaller Square from the Larger Square

To find the area of the shaded region, subtract the area of the smaller square from the area of the larger square. \[ \text{Area of shaded region} = (7x - 4)^2 - x^2 \]

Step 5: Expand and Simplify the Expression

Expand \( (7x - 4)^2 \) and simplify the expression. \[ (7x - 4)^2 = 49x^2 - 56x + 16 \] \[ \text{Area of shaded region} = 49x^2 - 56x + 16 - x^2 \] \[ \text{Area of shaded region} = 48x^2 - 56x + 16 \]

Final Answer

\[ \text{Area of the shaded region} = 48x^2 - 56x + 16 \]

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