Question

a. write a polynomial equation to represent the situation.
$\square x^2 + \square x + \square = 285$
part b
b. write and solve a system of equations.
$y = \square x^2 + \square x + \square$, $y = \square$; $x = \square$ ft
c. what are the dimensions of the garden plus border? list the dimensions from least to greatest.
$\square$ ft by $\square$ ft

Explanation:

Step1: Define total area equation

The total area of the garden plus border is given as 285 sq ft. The length of the total area is $16 + 2x$ ft, and the width is $12 + 2x$ ft. Expand the product of these dimensions:
$$(16+2x)(12+2x)=285$$
$$4x^2 + 56x + 198 = 285$$
$$4x^2 + 56x - 87 = 0$$

Step2: Set up system for Part B

Let $y$ represent the total area. The system is:
$y = 4x^2 + 56x + 198$
$y = 285$
Substitute $y=285$ into the first equation, then solve the quadratic $4x^2 + 56x - 87 = 0$ using the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=4$, $b=56$, $c=-87$:
$$x=\frac{-56\pm\sqrt{56^2-4(4)(-87)}}{2(4)}$$
$$x=\frac{-56\pm\sqrt{3136+1392}}{8}$$
$$x=\frac{-56\pm\sqrt{4528}}{8}$$
$$x=\frac{-56\pm 67.29}{8}$$
We take the positive root: $x=\frac{11.29}{8}\approx1.41$ (rounded to two decimals, or exact form $x=\frac{-28+\sqrt{1132}}{4}$)

Step3: Calculate total dimensions

Width: $12 + 2x = 12 + 2(1.41)=14.82$ ft
Length: $16 + 2x = 16 + 2(1.41)=18.82$ ft

Answer:

a. $4x^2 + 56x - 87 = 285$ (filled blanks: $4$, $56$, $-87$)
b. $y = 4x^2 + 56x + 198$; $y = 285$; $x\approx1.41$ ft
c. $14.82$ ft by $18.82$ ft

(Note: Using exact root $x=\frac{-28+\sqrt{1132}}{4}$, the dimensions are $12+2(\frac{-28+\sqrt{1132}}{4})=6-14+\frac{\sqrt{1132}}{2}=\frac{\sqrt{1132}}{2}-8$ and $16+2(\frac{-28+\sqrt{1132}}{4})=8-14+\frac{\sqrt{1132}}{2}=\frac{\sqrt{1132}}{2}-6$, which simplify to $\sqrt{283}-8$ and $\sqrt{283}-6$ since $\sqrt{1132}=2\sqrt{283}$)