Given a polynomial: $ P(x_1, x_2) = (ax_1+b)(cx_2+d)$
This can be written in another form as: $ P(x_1, x_2) = d_1x_1x_2 + d_2x_1 + d_3x_2 + d_4$
where, $d_1 = ac$, $d_2 = ad$, $d_3 = bc$, $d_4 = bd$
A feasible solution does not always exists for the coefficients $a,b,c \text{ and } d$, when $d_1,d_2,d_3 \text{ and } d_4$ are given. This can be seen by checking the condition: $d_1d_4=d_2d_3$.
But, what can be the intuitive explanation of the fact that a feasible solution does not always exists for transformation of two equivalent mathematical expressions?