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Fill in the blanks with positive integers:
(3 + sqrt(5))*3*(5 + 3*sqrt(5))^3 = ___ + ___* sqrt(5)

 Nov 9, 2023
 #2
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To make writing easier, replace √5 

by x in the left side of the original expression. After replacing, you will get this polynomial (3+x)*3*(5+3x))^3.
It is

(3 + x)*3*(5 + 3x))^3 = 81x^4 + 648x^3 + 1890x^2 + 2400x + 1125. (*)

Now, to get first integer number of the right side, I consider the terms of this polynomial (*) with even degrees

even(x) = 81x^4 + 1890x^2 + 1125

and substitute there x= 5.

I get then "second integer number in the answer" = 648*5 + 2400 = which is easy to compute = 5640. Therefore, the final ANSWER is 12600 + 5640* √5 (**)

Finally, to check the answer, I calculated (**) and compared with the left side of the original expression. I got the same number, which confirms correctness of my solution.

 Nov 11, 2023

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