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

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.

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so subtract the coefficients and retain the base and exponent:

-4b⁵ - 5b⁵
(-4 - 5)b⁵
-9b⁵

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{4}{9} \) x \( \frac{1}{6} \) = \( \frac{4 x 1}{9 x 6} \) = \( \frac{4}{54} \) = \(\frac{2}{27}\)

To multiply terms with radicals, multiply the coefficients and radicands separately:

3\( \sqrt{7} \) x 6\( \sqrt{9} \)
(3 x 6)\( \sqrt{7 \times 9} \)
18\( \sqrt{63} \)

Now we need to simplify the radical:

18\( \sqrt{63} \)
18\( \sqrt{7 \times 9} \)
18\( \sqrt{7 \times 3^2} \)
(18)(3)\( \sqrt{7} \)
54\( \sqrt{7} \)