a) \(\left(a-b\right)^2-2\left(a-b\right)c+c^2\)
\(=\left(a-b-c\right)^2\)
b) \(81a^2+18a+1\)
\(=\left(9a\right)^2+2.9a.1+1^2\)
\(=\left(9a+1\right)^2\)
c) \(3\left(x+1\right)^n.y-6\left(x+1\right)^{n+1}\)
\(=3\left(x+1\right)^n.y-6\left(x+1\right)^n.6\left(x+1\right)\)
\(=3\left(x+1\right)^n\left[y-12\left(x+1\right)\right]\)
\(=3\left(x+1\right)^n\left(y-12x-12\right)\)
d) \(\left(a-2b\right)^{3n}+\left(a-2b\right)^{3n+1}\)
\(=\left(a-2b\right)^{3n}+\left(a-2b\right)^{3n}.\left(a-2b\right)\)
\(=\left(a-2b\right)^{3n}\left(1+a-2b\right)\)