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\(A-1=\left(x+1\right)\left(x^2+1\right)...\left(x^{256}+1\right)\)
\(\Rightarrow\left(A-1\right)\left(x-1\right)=\left(x-1\right)\left(x+1\right)\left(x^2+1\right)...\left(x^{256}+1\right)\)
\(\Rightarrow\left(A-1\right)\left(x-1\right)=\left(x^2-1\right)\left(x^2+1\right)...\left(x^{256}+1\right)\)
\(\Rightarrow\left(A-1\right)\left(x-1\right)=\left(x^4-1\right)\left(x^4+1\right)...\left(x^{256}+1\right)\)
\(\Rightarrow\left(A-1\right)\left(x-1\right)=\left(x^{256}-1\right)\left(x^{256}+1\right)=x^{512}-1\)
\(\Rightarrow A-1=\dfrac{x^{512}-1}{x-1}\)
\(\Rightarrow A=\dfrac{x^{512}-1}{x-1}+1=\dfrac{x^{512}+x-2}{x-1}\)
\(=\dfrac{\left(x+2\right)^2}{x}\cdot\dfrac{x+2-x^2}{x+2}-\dfrac{x^2+6x+4}{x}\)
\(=\dfrac{\left(x+2\right)\left(-x^2+x+2\right)-x^2-6x-4}{x}\)
\(=\dfrac{-x^3+x^2+2x-2x^2+2x+4-x^2-6x-4}{x}\)
\(=\dfrac{-x^3-2x^2-2x}{x}=-x^2-2x-2\)
\(\left(x+2\right)\left(x-2\right)-\left(x+2\right)^2\)
\(=\left(x+2\right)\left(x-2-x-2\right)\)
\(=\left(-4\right)\left(x+2\right)\)
\(\left(x+1\right)^4-6\left(x+1\right)^2-\left(x^2-2\right)\left(x^2+2\right)\\ =x^4+4x^3+6x^2+4x+1-6x^2-12x-6-x^4+4\\ =4x^3-8x+5\)