Answer:
We know that:
(a - b)^2 = a^2 - 2ab + b^2
and:
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ac + 2bc + 2ab
Then:
(a - b)^4 = (a - b)^2*(a- b)^2 = (a^2 - 2ab + b^2)^2
Now we can just solve that last expression using the second expansion.
So, if we apply that to our case, we will have:
(1 - 4x)^2 = (1^2 - 2*1*4x + (4x)^2)^2
= (1 - 8x + 16x^2)^2
= 1^2 + (-8x)^2 + (16x^2)^2 + 2*1*(-8x) + 2*1*(16x^2) + 2*(-8x)*(16x^2)
= 1 + 64x^2 + 256x^4 - 16x + 32x^2 - 256x^3
= 256x^4 - 256x^3 + 96x^2 - 16x + 1
