jmc

When we expand we get $$\begin{array}{rl}& 3(x^2-x^3)+2(x-2x^2+3x^5)-(4x^3-x^2)\\ & =3x^2-3x^3+2x-4x^2+6x^5-4x^3+x^2\\ & =6x^5-7x^3+2x.\end{array}$$The coefficient of $x^2$ is $3-4+1=\boxed{0}$.