(1-3x)(1+2x)=1-x-6x^2=1-x(1+6x). (1-3x)^6(1+2x)^7=(1+2x)((1-3x)(1+2x))^6.
Using the binomial expansion of (...)^6 for (1-x(1+6x))^6:
(1-3x)^6(1+2x)^7=(1+2x)(1-x(1+6x))^6=(1+2x)(1-6x(1+6x)+15x^2(1+6x)^2-...).
We need to identify only which terms are going to include x^2 and x^3.
(1+2x)(1-6x-36x^2+15x^2(1+12x+36x^2)-20x^3(1+18x+108x^2+216x^3)+...)
...(-36+15-12)x^2+(-72+30+180-20)x^3+.... Therefore the coefficient of x^2 is -33 and x^3 is118.