We are tasked with finding a polynomial that represents the area of the border surrounding a flower bed. The flower bed's dimensions are $28 \, \text{ft}$ by $42 \, \text{ft}$, and it is surrounded by a uniform border of width $x$ feet.

Part (a) - Polynomial for the Area of the Border

1. Area of the entire rectangle (flower bed + border):

   • The total width of the rectangle (including the border on both sides) is $2x + 28$ feet.
   • The total length of the rectangle (including the border on both ends) is $2x + 42$ feet.
   • Thus, the area of the entire rectangle is given by the expression: $\text{Total Area} = (2x + 28)(2x + 42)$

2. Area of the flower bed (without the border):

   • The area of the flower bed is simply the product of its length and width: $\text{Area of flower bed} = 28 \times 42 = 1176 \, \text{square feet}$

3. Area of the border:

   • The area of the border is the total area minus the area of the flower bed: $\text{Area of border} = \text{Total Area} - \text{Area of flower bed}$ So, the polynomial for the area of the border is: $A(x) = (2x + 28)(2x + 42) - 1176$

Now, we can expand and simplify this expression:

• First, expand $(2x + 28)(2x + 42)$: $(2x + 28)(2x + 42) = 4x^2 + 84x + 56x + 1176 = 4x^2 + 140x + 1176$

• Subtract the area of the flower bed: $A(x) = (4x^2 + 140x + 1176) - 1176 = 4x^2 + 140x$

Thus, the polynomial that describes the area of the border is: $A(x) = 4x^2 + 140x$

Part (b) - Finding the Width of the Strip

We are told that 296 plants, each requiring 1 square foot of space, are to be planted in the border. Therefore, the area of the border is 296 square feet.

Using the polynomial from part (a), we set the area equal to 296: $4x^2 + 140x = 296$

Now, solve this quadratic equation:

1. Rearrange the equation: $4x^2 + 140x - 296 = 0$

2. Solve using the quadratic formula: The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $4x^2 + 140x - 296 = 0$, the coefficients are:

   • $a = 4$
   • $b = 140$
   • $c = -296$

   Substituting these values into the quadratic formula: $x = \frac{-140 \pm \sqrt{140^2 - 4(4)(-296)}}{2(4)}$ $x = \frac{-140 \pm \sqrt{19600 + 4736}}{8}$ $x = \frac{-140 \pm \sqrt{24336}}{8}$ $x = \frac{-140 \pm 156}{8}$

   Now, we have two possible solutions: $x = \frac{-140 + 156}{8} = \frac{16}{8} = 2 \, \text{feet}$ $x = \frac{-140 - 156}{8} = \frac{-296}{8} = -37$

Since $x$ must be positive, the width of the strip is $x = 2$ feet.

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Summary of Answers:

• (a) The polynomial for the area of the border is: $A(x) = 4x^2 + 140x$
• (b) The width of the strip is $2$ feet.

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Would you like to go into more details or ask any questions?

Follow-up questions:

1. How can we factor the quadratic expression in part (a)?
2. What would happen if the number of plants required was different from 296?
3. Can we solve the quadratic equation using a different method, like completing the square?
4. How would increasing the width of the strip affect the number of plants that can be planted?
5. How do the dimensions of the flower bed influence the complexity of the border?

Tip: When solving quadratic equations, it's always good to check if factoring or completing the square is possible before using the quadratic formula.