Since area of a rectangle is length x width, which we can abbreviate as A = L X W with A = area, L= length and W= width, then the width can be expressed as:
W = A/L
In our problem, W = (x² + 7x + 13)/(x + 4)
We can use polynomial long division to evaluate this. The result will be (x + 3) with a remainder of 1.
Therefore, the area A of this rectangle can be written as:
A = (x + 4)[(x + 3) + 1/(x + 4)]
Now, we know that the perimeter P of a rectangle is equal to 2L + 2W, with L = length and W = width of the rectangle.
P = 2L + 2W = 7 for the rectangle in the problem
P = 2(x + 4) + 2[(x + 3) + 1/(x + 4)] = 7
2x + 8 + 2x + 6 + 2/(x + 4) = 7
4x + 14 + 2/(x + 4) = 7
4x(x + 4) + 14(x + 4) + 2 = 7(x +4)
4x² + 16x + 14x + 56 + 2 = 7x + 28
4x² + 30x + 58 = 7x + 28
4x² + 23x + 30 = 0
(4x + 15)(x + 2) = 0
4x + 15 = 0
4x = -15
x = -15/4 = -3 3/4
and x + 2 = 0
x = -2
The only value of x that will not result in a negative dimension is x = -2
If x = -2, then L = -2 + 4 = 2 = the length of the rectangle
W = -2 + 3 + 1/(-2 + 4) = 1 + 1/2 = 3/2 = 1 1/2 = the width of the rectangle
