To find out how much greater the length of the field is than the width of the field, we need to subtract the expression that represents the width from the expression that represents the length.
The length of the field is given by:
[tex]\[ 14x - 3x^2 + 2y \][/tex]
The width of the field is given by:
[tex]\[ 5x - 7x^2 + 7y \][/tex]
We need to find the difference:
[tex]\[ \text{Difference} = (\text{Length}) - (\text{Width}) \][/tex]
Let's subtract the width expression from the length expression:
[tex]\[ (14x - 3x^2 + 2y) - (5x - 7x^2 + 7y) \][/tex]
First, distribute the negative sign across the second expression:
[tex]\[ 14x - 3x^2 + 2y - 5x + 7x^2 - 7y \][/tex]
Next, combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[ -3x^2 + 7x^2 = 4x^2 \][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[ 14x - 5x = 9x \][/tex]
- Combine the [tex]\(y\)[/tex] terms:
[tex]\[ 2y - 7y = -5y \][/tex]
Putting it all together, we get:
[tex]\[ 4x^2 + 9x - 5y \][/tex]
So, the difference between the length and the width of the field is:
[tex]\[ 4x^2 + 9x - 5y \][/tex]
Among the given options, this matches the answer:
[tex]\[ 4x^2 + 9x - 5y \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{4x^2 + 9x - 5y} \][/tex]
