\(a.\left(x-y\right)^{^{ }2}+\left(x+y\right)^{^{ }2}+\left(x-y\right)\left(x+y\right)\\ =x^{^2}-2xy+y^{^2}+x^{^2}+2xy+y^{^2}+x^{^2}-y^{^2}\\ =3x^{^2}+y^{^2}\\ b.x\left(x-1\right)\left(x+1\right)-\left(x+1\right)^{^3}\\ =x\left(x^{^2}-1\right)-\left(x+1\right)^{^3}\\ =x^{^3}-x-x^{^3}-3x^{^2}-3x-1\\ =-3x^{^2}-4x-1\)
\(c.\left(5a+\dfrac{1}{2}\right)^{^2}-2\left(25a^{^2}-\dfrac{1}{4}\right)+\left(5a-\dfrac{1}{2}\right)^{^2}\\ =\left(5a+\dfrac{1}{2}\right)^{^2}-2\left(5a-\dfrac{1}{2}\right)\left(5a+\dfrac{1}{2}\right)+\left(5a-\dfrac{1}{2}\right)^{^2}\\ =\left(5a+\dfrac{1}{2}-5a+\dfrac{1}{2}\right)=1\\ d.\left(a+b\right)^{^3}-\left(a-b\right)^{^3}-6a^{^2}b\\ =a^{^3}+3a^{^2}b+3ab^{^2}+b^{^3}-\left(a^{^3}-3a^{^2}b+3ab^{^2}-b^{^3}\right)-6a^{^2}b\\ =a^{^3}+3a^{^2}b+3ab^{^2}+b^{^3}-a^{^3}+3a^{^2}b-3ab^{^2}+b^{^3}-6a^{^2}b\\ =2b^{^3}\)
