Hello, I am puzzled about this operation (1) with polynomials in a general proof of Exponential input theorem and I don't understand the the concept ofabeing s-fold zero.

p(D) = q(D)(D - a)^{s }q(a) = nonzeroThe above implies

The proof lets k be the degree of q(D) and writes it in powers of (D-a)

p^{(s)}(a) = q(a)s!

(1) q(D) = q(a) + c_{1}(D - a) + ... c_{k}(D - a)^{k }

(2) q(D) =q(a)(D−a)then substitute D with a^{s}+c_{1}(D−a)^{s+1 }+...+c_{k}(D−a)^{s+k }^{ }How did they figure (1) out?

if k = 2, I getI feel totally confused and as if I missed a lot

D^{2}+ c_{1}D +c_{2}= A^{2}+ c_{1}A + c_{2}+ c_{1}D - c_{1}A + c_{2}D^{2 }- c_{2}2DA + c_{2}A^{2 }

Could you please point out what I do not understand?

Thank you