a) \(\left(x+1\right)\left(x^3-x+1\right)\)
\(=\left(x+1\right)\left(x^2-1\cdot x+1^2\right)\)
\(=x^3+1^3\)
\(=x^3+1\)
b) \(\left(6x+1\right)\left(36x^2-6x+1\right)\)
\(=\left(6x+1\right)\left[\left(6x\right)^2-6x\cdot1+1^2\right]\)
\(=\left(6x\right)^3+1^3\)
\(=216x^3+1\)
c) \(\left(3x+2\right)\left(9x^2-6x+4\right)\)
\(=\left(3x+2\right)\left[\left(3x\right)^2-3x\cdot2+2^2\right]\)
\(=\left(3x\right)^3+2^3\)
\(=27x^3+8\)
d) \(\left(2x+5y\right)\left(4x^2-10xy+25y^2\right)-\left(x-y\right)\left(x^2+xy+y^2\right)\)
\(=\left(2x+5y\right)\left[\left(2x\right)^2-2x\cdot5y+\left(5y^2\right)\right]-\left(x^3-y^3\right)\)
\(=\left[\left(2x\right)^3+\left(5y\right)^3\right]-\left(x^3-y^3\right)\)
\(=8x^3+125y^3-x^3+y^3\)
\(=7x^3+126y^3\)
1: (x+1)(x^2-x+1)=x^3+1
2: (6x+1)(36x^2-6x+1)
=(6x)^3+1^3=216x^3+1
3: (3x+2)(9x^2-6x+4)
=(3x)^3+2^3
=27x^3+8
4: =8x^3+125y^3-x^3+y^3
=7x^3+126y^3
