Official solution
Given,
(2 - x2) . (1 + 2x + 3x2) 6 + (1 - 4x2)6)
Let, a = ((1 + 2x + 3x2)6 + (1 - 4x2)6)
\[\therefore\,\,\,\,\] Given statement becomes,
(2 - x2) . (a)
= 2a - x2 (a)
Here coefficients of x2 is
= 2 (coefficient of x2 in a ) - 1 (constant in a)
(1 + 2x + 3x2) 6 = 6C0 + 6C1 (2x + 3x2) + 6C2 (2x + 3x2)2
+ . . . . . .+ (2x + 3x2)6
(1 - 4x2)6 = 6C0 - 6C1 (4x2) + 6C2 (4x2)2
+ . . . . .+ (4x2)6
Coefficient of x2 in (1 + 2x + 3x2)6
= 6C1 \[ \times \] 3 + 6C2 \[ \times \] 4
= 18 + 60
Coefficient of x2 in (1 - 4x2)6
= - 6C1 \[ \times \] 4
= - 24
Coefficient of x2 in ((1 + 2x + 3x2)6 + (1 - 4x2)6)
= 60 + 18 - 24
= 54
Constant term in (1 + 2x + 3x2)6 = 6C0 = 1
Constant term in (1 - 4x)6 = 6C0 = 1
\[\therefore\,\,\,\,\] Constant term in ((1 + 2x + 3x2)6 + (1 - 4x)6)
= 1 + 1 = 2
\[\therefore\,\,\,\,\] Coefficient of x2 in (2 - x2) ((1 + 2x + 3x2)6 + (1 - 4x2)6)
= 2 (54) - 1 (2)
= 108 - 2
= 106