We know that, x + y + z 2 = x 2 + y 2 + z 2 + 2 x y + 2 y z + 2 z x Here, x = a , y = b , z = - c Hence, a + b - c 2 = a 2 + b 2 + c 2 + 2 a b - 2 b c - 2 c a

We know that, x + y + z 2 = x 2 + y 2 + z 2 + 2 x y + 2 y z + 2 z x Here, x = a , y = - b , z = c Hence, a - b + c 2 = a 2 + b 2 + c 2 - 2 a b - 2 b c + 2 c a

We are given to expand 3 x + 1 3 x 2 . Thus, we have, 3 x + 1 3 x 2 = 3 x 2 + 2 × 3 x × 1 3 x + 1 3 x 2 [ ∵ a + b 2 = a 2 + 2 a b + b 2 = 9 x 2 + 2 + 1 9 x 2 = 9 x 2 + 1 9 x 2 + 2 Therefore, the required 9 x 2 + 1 9 x 2 + 2

We are given to expand 2 x - 1 2 x 2 . Thus, we have, 2 x - 1 2 x 2 = 2 x 2 - 2 × 2 x × 1 2 x + 1 2 x 2 [ ∵ a - b 2 = a 2 - 2 a b + b 2 = 4 x 2 - 2 + 1 4 x 2 Therefore, the required answer is 4 x 2 + 1 4 x 2 - 2 .