The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

In order to find how many additional workers are needed to staff the extra crews you first need to calculate how many workers are on a crew. There are 12 workers at the company now and that's enough to staff 3 crews so there are \( \frac{12}{3} \) = 4 workers on a crew. 6 crews are needed for the busy season which, at 4 workers per crew, means that the roofing company will need 6 x 4 = 24 total workers to staff the crews during the busy season. The company already employs 12 workers so they need to add 24 - 12 = 12 new staff for the busy season.

The equation for this sequence is:

a_n = a_n-1 + 4(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 4(6 - 1)
a₆ = 41 + 4(5)
a₆ = 61

A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 12 meters so the equation becomes: 2w + 2h = 12.

Putting these two equations together and solving for width (w):

2w + 2h = 12
w + h = \( \frac{12}{2} \)
w + h = 6
w = 6 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 6 - 2w
3w = 6
w = \( \frac{6}{3} \)
w = 2

Since h = 2w that makes h = (2 x 2) = 4 and the area = h x w = 2 x 4 = 8 m²