We are doing solving polynomials using the zero factor property and here is the question:

Marie has a rectangular board 18 inches by 23 inches around which she wants to put a uniform border of shells. If she has enough shells for a border whose area is 282 square inches, determine the width of the border. (the border has x's as the width and length)

So the board AND the border have an area of

A2=(18+x)(23+x)
The board alone is
A1=18*23
A2-A1=282
Solve for x.

To solve this question, we can start by finding the area of the rectangular board and subtracting it from the area of the board plus the border. This will give us the area of the border itself.

The formula for the area of a rectangle is length multiplied by width. So, the area of the rectangular board is 18 inches by 23 inches, which gives us 18 * 23 = 414 square inches.

Now we need to find the area of the board plus the border. We are given that the area of the border is 282 square inches. Let's represent the width of the border as 'x'.

So, the area of the board plus the border is (18 + 2x) inches by (23 + 2x) inches, which can be expressed as (18 + 2x) * (23 + 2x) square inches.

Now, we can set up an equation to represent the problem:

(18 + 2x) * (23 + 2x) - 414 = 282

Simplifying the equation:

(18 + 2x) * (23 + 2x) = 696

Expanding the equation:

414 + 36x + 46x + 4x^2 = 696

Rearranging and combining like terms:

4x^2 + 82x - 282 = 0

We now have a quadratic equation. To solve for x, we can use factoring, completing the square, or the quadratic formula. In this case, let's use factoring.

We need to find two numbers that multiply to give us -282 and add up to 82. After some trial and error, we find that the numbers are 14 and 4.

So, we can rewrite our equation as:

(2x + 14)(2x - 4) = 0

Now we can set each factor equal to zero and solve for x:

2x + 14 = 0 or 2x - 4 = 0

Solving each equation separately:

2x + 14 = 0 leads to x = -7
2x - 4 = 0 leads to x = 2

Since we are dealing with the width of the border, we can disregard the negative value of x. Therefore, the width of the border is 2 inches.

Therefore, Marie should put a uniform border of shells with a width of 2 inches around the rectangular board.