To find the correct equation for the increase in side length of the square, we start by determining the side length after the increase.
The original side length of the square is 4 inches. If we let \( x \) represent the increase in side length, the new side length of the square will be \( 4 + x \).
The area of a square is given by the square of its side length, so the area of the enlarged square can be expressed as:
\[ (4 + x)^2 \]
We are given that this area should be equal to 81 square inches. Therefore, we can set up the equation:
\[ (4 + x)^2 = 81 \]
Next, we can expand the left side:
\[ (4 + x)^2 = 16 + 8x + x^2 \]
So the equation becomes:
\[ 16 + 8x + x^2 = 81 \]
To simplify this, we subtract 81 from both sides:
\[ x^2 + 8x + 16 - 81 = 0 \]
\[ x^2 + 8x - 65 = 0 \]
Thus, the equation that can be used to solve for \( x \), the increase in the side length of the square in inches, is:
\[ x^2 + 8x - 65 = 0 \]
So the correct option is:
x² + 8x - 65 = 0.
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