I have calculated several homogeneous polynomials in 4,5 or 6 variables $t_1,\dots,t_6$.
I would like to rewrite them as a sum of products of specific lower degree polynomials, which have a meaning in the problem I'm solving.
For example, the simplest polynomial I have is $$ P_4(t_1,t_2,t_3,t_4) = t_1t_2+t_1(t_3+t_4)+t_2(t_3+t_4) $$ which I would like to express in terms of $t_1+t_2, t_2+t_3+t_4, t_1+t_3+t_4$ (only them, although not necessarily all of them). For example, it's easy to see that $$P_4(t_1,t_2,t_3,t_4) = (t_1+t_2)(t_2+t_3+t_4) - t_2^2$$ but I'd prefer something like $$P_4(t_1,t_2,t_3,t_4) = a(t_1+t_2)(t_2+t_3+t_4) + b(t_1+t_2)(t_1+t_3+t_4) + c(t_1+t_3+t_4)(t_2+t_3+t_4)$$
Question: Is there a way to ask Mathematica this sort of partial factorization?
Thank you in advance!
Note: I don't know if a solution exists for my exact polynomials (maybe I have to search for a different interpretation), but I think the general problem is interesting in its own right.