Just two cases Possible here :
1) Extract x from (1+X)^{3 }and x^{2 }from (2+x^{2})^{10 }to finally get x^{3}. --> ^{3}C_{1 }* ^{10}C_{1 }* 2^{9}
2) Extract x^{3} from (1+X)^{3 }and x^{0 }from (2+x^{2})^{10 }to finally get x^{3}. --> ^{3}C_{3 }* ^{10}C_{0 }* 2^{10}
So coefficient of x^{3} is --> case 1 + case 2 --> 30*2^{9} + 2^{10 }--> 32 * 2^{9 }--> 2^{14}.