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Your question is confusingly worded. As far as I can tell, you have a polynomial in several variables, and want to reduce the power on all of a given variable by 1, and remove the terms which are constant with respect to that variable. Here is a way to do that:
syms a b c d
f=a^2*b+b^2*c+c*d
[p,q] = coeffs(f,c)
% p = [ b^2 + d, a^2*b]
% q = [ c, 1]
sum(p(1:end-1).*q(2:end))
coeffs returns the coefficients of the various powers of c in the expression f, returning the coefficients in p and the powers of c in q. Then multiply the i-th entry of p by the I+1-th entry of q and add them up.