I'm not sure what you mean...did you mean to say that when you're finding an eigenvector by solving , you didn't get the vector v=[3,-2,1]? That's all right, because there are actually an infinite amount of eigenvectors, and they are precisely the set where v=[3,-2,1]. So as long as you got a multiple of v, you're fine.

But judging from the question, all you had to do was left multiply your vector v by the matrix A and verify that Av=[-15,10,-5]=-5[3,-2,1]=-5v.