The possible values of (0, ) such that + (4 )+ (7 )=0 are

Step-by-step solution

Given + (4 ) + (7 ) = 0 , group the outer terms and apply the sum-to-product identity: 2 ( + 7 2 ) ( - 7 2 ) + (4 ) = 2 (4 ) (3 ) + (4 ) Factor (4 ) : (4 )(2 (3 ) + 1) = 0 Either (4 ) = 0 or 2 (3 ) + 1 = 0 . For (4 ) = 0 , = n 4 ; in (0, ) , = 4 , 2 , 3 4 . For (3 ) = - 1 2 , 3 = 2k 2 3 , so = 2k 3 2 9 . In (0, ) , = 4 9 , 2 9 , 8 9 . Combining and ordering gives 2 9 , 4 , 4 9 , 2 , 3 4 , 8 9 , corresponding to option D .