Find the expansions of (1-x)^6 and (1+2x)^6 as far as the terms in x^3. Hence expand (1+x-2x^2)^6 up to the terms in x^3.
Using the exponent rule
a^n * b^n = (ab)^n
(1-x)^6 (1+2x)^6
= (1 + x - 2x^2)^6
but (1-x)^6 = 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6)
and
(1+2x)^6 = 1 + 12x + 60x^2 + 160x^3 + ..)
now considering only those multiplications that result in x^3 ....
(1 + x - 2x^2)^6
=(1-x)^6 (1+2x)^6
= (1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6)(1 + 12x + 60x^2 + 160x^3 + ..)
= ... + 160x^3 + ... (-6x)(60x^2) ... + (-20x^3(1) + ..
collecting the x^3 terms
160x^3 -360x^3 +180x^3 - 20x^3
=-40x^3
check my arithmetic, ( I should write it down on paper first)