The coefficient of x^2 in the expansion of (1 - 2x^2)^2 (x + 2 x )^8 is equal to

Step-by-step solution

The given expression is (1 - 2x^2)^2 (x + 2 x )^8 . Expanding the first term, we get (1 - 4x^2 + 4x^4) . The general term in the expansion of (x + 2 x )^8 is: T_ r+1 = ⁸C_ r x^ 8-r ( 2 x )^r = ⁸C_ r 2^r x^ 8-2r To find the coefficient of x^2 in the product (1 - 4x^2 + 4x^4) ( ⁸C_ r 2^r x^ 8-2r ) , we need to collect terms that result in x^2 : 1) 1 ( term with x^2) : 8 - 2r = 2 r = 3 Coefficient = 1 ⁸C₃ 2^3 = 1 56 8 = 448 2) -4x^2 ( term with x^0) : 8 - 2r = 0 r = 4 Coefficient = -4 ⁸C₄ 2^4 = -4 70 16 = -4480 3) 4x^