a) \(\left(x+y\right)^2+\left(x-y\right)^2-2x^2=x^2+2xy+y^2+x^2-2xy+y^2-2x^2\)
\(=2x^2+2y^2-2x^2=2y^2\)
b) \(\left(x+1\right)^3-\left(x-1\right)\left(x^2+x+1\right)-3x\left(x+1\right)\)
\(=x^3+3x+3x^2+1-x^3+1-3x^2-3x\)
\(=1+1=2\)
c) \(\left(x^2+\dfrac{1}{3}x+\dfrac{1}{9}\right)\left(x-\dfrac{1}{3}\right)+\left(x-\dfrac{1}{3}\right)^2-x^3-x^2\)
\(=\left(x^3-\dfrac{1}{27}\right)+x^2-\dfrac{2}{3}x+\dfrac{1}{9}-x^3-x^2\)
\(=-\dfrac{2}{3}x+\dfrac{2}{27}\)
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