Maybe adding a term will help. Make the expression [(2 + √5)^100 + (2 - √5)^100] * [(1 + √2)^100 + (1 + √2)^-100] and see how much the (2-√5) affects the result.
You already noted the relation between 1+√2 and 1-√2. Now do that to 2+√5 and 2-√5. Also, 1-√2 and 2-√5 have an absolute value less than 1.
re: poor explanation
