Transforming the ternary cubic polynomial in to a quaternion cubic homogeneous polynomial Form

Question

I have a ternary cubic polynomial of the form $$ P(X,Y,Z) = \sum_{i=0}^{m1}\sum_{j=0}^{m2}\sum_{k=0}^{m3} c_{i,j,k}X^iY^jZ^k$$ how would one write it in quaternion from
i know the the answer should be $$ f(x_1,x_2,x_3,x_4) = \sum (a_i a_j a_k)x_ix_jx_k \ \ where \ x_1=1 $$ but not how to get from one form to the other or why it is correct