7 A farmer has a rectangular field whose area is \( \left(x^{2}+5 x+6\right) \mathrm{m}^{2} \) and width is \( (x+2) \mathrm{m} \). (a) What is the length of the ground? (b) Find the expanded form of \( (x+3)^{2} \) and explain what does it represent in the field?

To find the length of the field, first use the formula for the area of a rectangle: \( \text{Area} = \text{length} \times \text{width} \). Since the area is given by \( (x^{2}+5x+6) \) and the width by \( (x+2) \), we can set up the equation: \[ \text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{x^{2}+5x+6}{x+2} \] Upon performing polynomial long division or factoring, we see that: \[ (x^2 + 5x + 6) = (x + 2)(x + 3) \] Thus the length of the ground is \( (x + 3) \) meters. For part (b), the expanded form of \( (x + 3)^{2} \) is \( x^{2} + 6x + 9 \). In the context of the field, this represents the scenario where the length of the ground is doubled, possibly when adjusting dimensions for an expansion or redesign. Moreover, it gives us insight into how changing the length affects the area: a larger length implies a more extensive field or crops flourishing within a more generous space.