If Gary has seven suits, he can choose which of the seven he will place first. He can choose from the remaining six which one he will place next. This goes on until all the suits are placed in his wardrobe. In short, he can arrange his suits in 7! (7 factorial) ways in his wardrobe.
7! = 5040
Thus, Gary can arrange his suits in 5040 ways.
Factorization of the given polynomial to find the product is as follows;
Factorizing;
![\dfrac{x^2 + 3*x +2}{x^2+ 6*x + 5} = \dfrac{(x + 1) * (x + 2)}{(x + 1)*(x + 5)}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%5E2%20%2B%203%2Ax%20%2B2%7D%7Bx%5E2%2B%206%2Ax%20%2B%205%7D%20%3D%20%5Cdfrac%7B%28x%20%2B%201%29%20%2A%20%28x%20%2B%202%29%7D%7B%28x%20%2B%201%29%2A%28x%20%2B%205%29%7D)
![\dfrac{x^2 + 7*x +10}{x^2+ 4*x + 4} = \dfrac{(x + 2) * (x + 5)}{(x + 2)*(x + 2)}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%5E2%20%2B%207%2Ax%20%2B10%7D%7Bx%5E2%2B%204%2Ax%20%2B%204%7D%20%3D%20%5Cdfrac%7B%28x%20%2B%202%29%20%2A%20%28x%20%2B%205%29%7D%7B%28x%20%2B%202%29%2A%28x%20%2B%202%29%7D)
Expressing the product in terms of the factors
![\dfrac{x^2 + 3*x +2}{x^2+ 6*x + 5} \times \dfrac{x^2 + 7*x +10}{x^2+ 4*x + 4} = \dfrac{(x + 2) * (x + 5)}{(x + 2)*(x + 2)} \times \dfrac{(x + 1) * (x + 2)}{(x + 1)*(x + 5)}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%5E2%20%2B%203%2Ax%20%2B2%7D%7Bx%5E2%2B%206%2Ax%20%2B%205%7D%20%5Ctimes%20%20%5Cdfrac%7Bx%5E2%20%2B%207%2Ax%20%2B10%7D%7Bx%5E2%2B%204%2Ax%20%2B%204%7D%20%3D%20%5Cdfrac%7B%28x%20%2B%202%29%20%2A%20%28x%20%2B%205%29%7D%7B%28x%20%2B%202%29%2A%28x%20%2B%202%29%7D%20%5Ctimes%20%5Cdfrac%7B%28x%20%2B%201%29%20%2A%20%28x%20%2B%202%29%7D%7B%28x%20%2B%201%29%2A%28x%20%2B%205%29%7D)
The steps arranged in the order in which they would be performed are;
First step;
![\dfrac{x^2 + 3*x +2}{x^2+ 6*x + 5} \times \dfrac{x^2 + 7*x +10}{x^2+ 4*x + 4}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%5E2%20%2B%203%2Ax%20%2B2%7D%7Bx%5E2%2B%206%2Ax%20%2B%205%7D%20%5Ctimes%20%20%5Cdfrac%7Bx%5E2%20%2B%207%2Ax%20%2B10%7D%7Bx%5E2%2B%204%2Ax%20%2B%204%7D)
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Second step (factorizing)
![\dfrac{(x + 2) * (x + 5)}{(x + 2)*(x + 2)} \times \dfrac{(x + 1) * (x + 2)}{(x + 1)*(x + 5)}](https://tex.z-dn.net/?f=%5Cdfrac%7B%28x%20%2B%202%29%20%2A%20%28x%20%2B%205%29%7D%7B%28x%20%2B%202%29%2A%28x%20%2B%202%29%7D%20%5Ctimes%20%5Cdfrac%7B%28x%20%2B%201%29%20%2A%20%28x%20%2B%202%29%7D%7B%28x%20%2B%201%29%2A%28x%20%2B%205%29%7D)
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Third step (dividing out common terms)
![\dfrac{(x + 5)}{(x + 2)} \times \dfrac{(x + 2)}{(x + 5)}](https://tex.z-dn.net/?f=%5Cdfrac%7B%28x%20%2B%205%29%7D%7B%28x%20%2B%202%29%7D%20%5Ctimes%20%5Cdfrac%7B%28x%20%2B%202%29%7D%7B%28x%20%2B%205%29%7D)
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Fourth step (rearranging and removing terms that cancel each other)
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Learn more here:
brainly.com/question/12723898
Hey there! Is there more text to this? I would love to help, but there is no question.