To find the width of the rectangular field, we can use polynomial long division to divide the area polynomial \( x^3 + x^2 - 14x - 24 \) by the length polynomial \( x - 4 \).
We start with the long division:
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Divide the leading term of the area polynomial \( x^3 \) by the leading term of the length polynomial \( x \) to get \( x^2 \).
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Multiply \( x^2 \) by \( x - 4 \) to get \( x^3 - 4x^2 \).
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Subtract this from the original polynomial: \[ (x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = 5x^2 - 14x - 24 \]
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Now repeat the process with \( 5x^2 \):
- Divide \( 5x^2 \) by \( x \) to get \( 5x \).
- Multiply \( 5x \) by \( x - 4 \) to get \( 5x^2 - 20x \).
- Subtract: \[ (5x^2 - 14x - 24) - (5x^2 - 20x) = 6x - 24 \]
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Finally, take \( 6x \):
- Divide \( 6x \) by \( x \) to get \( 6 \).
- Multiply \( 6 \) by \( x - 4 \) to get \( 6x - 24 \).
- Subtract: \[ (6x - 24) - (6x - 24) = 0 \]
The result of the division is:
\[ x^2 + 5x + 6 \]
Therefore, the expression that represents the width of the field is:
\[ \boxed{x^2 + 5x + 6} \]
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