1: \(3\left(x-y\right)^2-2\left(x+y\right)^2-\left(x-y\right)\left(x+y\right)\)
\(=3\left(x^2-2xy+y^2\right)-2\left(x^2+2xy+y^2\right)-\left(x^2-y^2\right)\)
\(=3x^2-6xy+3y^2-2x^2-4xy-2y^2-x^2+y^2\)
\(=2y^2-10xy\)
2: \(\left(x-1\right)^2-2\left(x-1\right)\left(x-3\right)+\left(x-3\right)^2\)
\(=\left[\left(x-1\right)-\left(x-3\right)\right]^2\)
\(=\left(x-1-x+3\right)^2=2^2=4\)
3: \(\left(x-2\right)\left(x+2\right)\left(x^2+4\right)-\left(x^2-3\right)\left(x^2+3\right)\)
\(=\left(x^2-4\right)\left(x^2+4\right)-\left(x^2-3\right)\left(x^2+3\right)\)
\(=x^4-16-\left(x^4-9\right)\)
\(=x^4-16-x^4+9=-7\)
4: \(\left(x-2\right)^3-\left(x+1\right)\left(x^2+x+1\right)+6\left(x-1\right)^2\)
\(=x^3-6x^2+12x-8-\left(x^3+1\right)+6\left(x^2-2x+1\right)\)
\(=x^3-6x^2+12x-8-x^3-1+6x^2-12x+6\)
\(=-3\)
5: \(\left(6x+1\right)^2+\left(6x-1\right)^2-2\left(1+6x\right)\left(6x-1\right)\)
\(=\left(6x+1\right)^2-2\left(6x+1\right)\left(6x-1\right)+\left(6x-1\right)^2\)
\(=\left[\left(6x+1\right)-\left(6x-1\right)\right]^2\)
\(=\left(6x+1-6x+1\right)^2=2^2=4\)
6: \(12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot24\cdot\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot\left(5^2-1\right)\cdot\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot\left(5^{16}-1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot\left(5^{32}-1\right)\left(5^{32}+1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot\left(5^{64}-1\right)\left(5^{64}+1\right)\)
\(=\dfrac{1}{2}\cdot\left(5^{128}-1\right)\)
