Multinomial

An $x^6$ term can be formed from the following products in the expansion:

$(2x^2)^3(1)^7=8x^6$ . This occurs ${10\choose3}=120$ times in the expansion.

$(2x^2)^2(-x)^2(1)^6=4x^6$ . This occurs ${10\choose2}{8\choose2}=1260$ times in the expansion.

$(2x^2)^1(-x)^4(1)^5=2x^6$ . This occurs ${10\choose1}{9\choose4}=1260$ times in the expansion.

$(-x)^6(1)^4=x^6$ . This occurs ${10\choose6}=210$ times in the expansion.

Thus, the coefficient of the $x^6$ term is $8\times120+4\times1260+2\times1260+210=\boxed{8730}$ .