\(=\left(x+3\right)^6-y^6\\ =\left[\left(x+3\right)^3-y^3\right]\left[\left(x+3\right)^3+y^3\right]\\ =\left(x+3-y\right)\left[\left(x+3\right)^2+y\left(x+3\right)+y^2\right]\left(x+3+y\right)\left[\left(x+3\right)^2-y\left(x+3\right)+y^2\right]\\ =\left(x+y+3\right)\left(x-y+3\right)\left(x^2+6x+9+xy+3y+y^2\right)\left(x^2+6x+9-xy-3y+y^2\right)\)

\(\left(x^2+6x+9\right)^3-\left(y^2\right)^3=\left(x^2+6x+9-y^2\right)\left[\left(x^2+6x+9\right)^2+\left(x^2+6x+9\right)y^2+y^4\right]\)

\(=\left[\left(x+3\right)^2-y^2\right]\left\{\left[\left(x^2+6x+9\right)^2+2\left(x^2+6x+9\right)y^2+y^4\right]-\left(x^2+6x+9\right)y^2\right\}\)

\(=\left(x+3-y\right)\left(x+3+y\right)\left[\left(x^2+6x+9+y^2\right)^2-\left(x+3\right)^2y^2\right]\)

\(=\left(x+3-y\right)\left(x+3+y\right)\left[\left(x^2+6x+9+y^2\right)-\left(x+3\right)y\right]\left(x^2+6x+9+y^2\right)+\left(x+3\right)y\)

\(=\left(x+3-y\right)\left(x+3+y\right)\left(x^2+6x+9+y^2-xy-3y\right)\left(x^2+6x+9+y^2+xy+3y\right)\)

\(x^2\left(x+4\right)^2-\left(x+4\right)^2-\left(x^2-1\right)\\ =\left(x+4\right)^2\left(x^2-1\right)-\left(x^2-1\right)\\ =\left(x^2-1\right)\left[\left(x+4\right)^2-1\right]\\ =\left(x-1\right)\left(x+1\right)\left(x+4-1\right)\left(x+4+1\right)\\ =\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+5\right)\)

\(= (x+4)^2(x^2-1)-(x^2-1)=[(x+4)^2-1](x^2-1)\)

\(=(x+4-1)(x+4+1)(x-1)(x+1)\)

\(=(x+3)(x+5)(x-1)(x+1)\)

\(x^2\left(x+4\right)^2-\left(x+4\right)^2-\left(x^2-1\right)\)

\(=\left(x+4\right)^2\left(x^2-1\right)-\left(x^2-1\right)\)

\(=\left(x^2-1\right)\left[\left(x+4\right)^2-1\right]\)

\(=\left(x^2-1\right)\left(x+3\right)\left(x+5\right)\)

Ta có: (x²+6x-5)(x²+6x+3)-20

      = [(x²+6x-1)-4][(x²+6x-1)+4]-20

      = (x²+6x-1)²-16-20

      = (x²+6x-1)²-36

      = (x²+6x-7)(x²+6x-5)

      = (x+7)(x-1)(x²+6x-5)

\(\left(x^2+6x-5\right)\left(x^2+6x+3\right)\\ =\left(x^2+6x-1\right)^2-16-20\\ =\left(x^2+6x-1\right)^2-36\\ =\left(x^2+6x-1-6\right)\left(x^2+6x-1+6\right)\\ =\left(x^2+6x-7\right)\left(x^2+6x+5\right)\\ =\left(x-1\right)\left(x+7\right)\left(x+1\right)\left(x+5\right)\)

\(\left(x^2+6x-5\right)\left(x^2+6x+3\right)-20\)

\(=\left(x^2+6x\right)^2-2\left(x^2+6x\right)-35\)

\(=\left(x^2+6x-7\right)\left(x^2+6x+5\right)\)

\(=\left(x+7\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)\)

\(\left(x^2+5x-3\right)\left(x^2+5x-5\right)-15=\left(x^2+5x-3\right)\left(x^2+5x-3-2\right)-15=\left(x^2+5x-3\right)^2-2\left(x^2+5x-3\right)+1-16=\left(x^2+5x-3-1\right)^2-4^2=\left(x^2+5x-4\right)^2-4^2=\left(x^2+5x-8\right)\left(x^2+5x\right)=x\left(x+5\right)\left(x^2+5x-8\right)\)

\(\left(x^2+5x-3\right)\left(x^2+5x-5\right)-15\)

\(=\left(x^2+5x\right)^2-8\left(x^2+5x\right)-15\)

\(=x\left(x+5\right)\left(x^2+5x-8\right)\)

\(x^2+3y^2-4xy+10x-12y+9\)

\(=\left(x^2-xy+x\right)+9x-3xy+3y^2-12y+9\)

\(=\left(x^2-xy+x\right)+\left(9x-9y+9\right)-3xy+3y^2-3y\)

\(=\left(x^2-xy+x\right)+\left(9x-9y+9\right)-\left(3xy-3y^2+3y\right)\)

\(=x\left(x-y+1\right)+9\left(x-y+1\right)-3y\left(x-y+1\right)\\ =\left(x-y+1\right)\left(x+9-3y\right)\)

\(x^2-x-2020.2021=x^2+2020x-2021x-2020.2021=x\left(x+2020\right)-2021\left(x+2020\right)=\left(x+2020\right)\left(x-2021\right)\)

\(x^2-x-2020\cdot2021\)

\(=\left(x-2021\right)\left(x+2020\right)\)

\(\left(x^2+x\right)^2+4x^2+4x-12=\left[\left(x^2+x\right)^2+4\left(x^2+x\right)+4\right]-16=\left(x^2+x+2\right)-4^2=\left(x^2+x+2-4\right)\left(x^2+x+2+4\right)=\left(x^2+x-2\right)\left(x^2+x+6\right)=\left(x-1\right)\left(x+2\right)\left(x^2+x+6\right)\)

\(\left(x^2+x\right)^2+4x^2+4x-12\\ =\left(x^2+x+2\right)-4\\ =\left(x^2+x-2\right)\left(x^2+x+6\right)\)

\(\left(x^2+x\right)^2+4x^2+4x-12\)

\(=\left(x^2+x+6\right)\left(x^2+x-2\right)\)

\(=\left(x^2+x+6\right)\left(x+2\right)\left(x-1\right)\)

\(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2=\left(x^2+4x+8\right)^2+2x\left(x^2+4x+8\right)+x\left(x^2+4x+8\right)+2x^2\)

\(=\left(x^2+4x+8\right)\left(x^2+4x+8+2x\right)+x\left(x^2+4x+8+2x\right)\)

\(=\left(x^2+4x+8\right)\left(x^2+6x+8\right)+x\left(x^2+6x+8\right)\)

\(=\left(x^2+4x+8+x\right)\left(x^2+6x+8\right)=\left(x^2+5x+8\right)\left(x^2+6x+8\right)\)

Ta có: \(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2\)

\(=\left(x^2+5x+8\right)\left(x^2+6x+8\right)\)

\(=\left(x^2+5x+8\right)\left(x+2\right)\left(x+4\right)\)

\(x^{n+3}+x^n=x^n.x^3+x^n=x^n\left(x^3+1\right)=x^n\left(x+1\right)\left(x^2-x+1\right)\)

\(x^{n+3}+x^n=x^n\left(x^3+1\right)=x^n\left(x+1\right)\left(x^2-x+1\right)\)

\(x^{n+3}+x^n=x^n\left(x^3+1\right)=x^n\cdot\left(x+1\right)\left(x^2-x+1\right)\)

x²-2x-15=(x²-5x)+(3x-15)=x(x-5)+3(x-5)=(x-5)(x+3)

\(x^2-3x-15=\left(x^2-2.\dfrac{3}{2}x+\dfrac{9}{4}\right)-\dfrac{69}{4}=\left(x-\dfrac{3}{2}\right)^2-\left(\dfrac{\sqrt{69}}{2}\right)^2\)

\(=\left(x-\dfrac{3}{2}-\dfrac{\sqrt{69}}{2}\right)\left(x-\dfrac{3}{2}+\dfrac{\sqrt{69}}{2}\right)\)

\(x^2-2x-15=\left(x-5\right)\left(x+3\right)\)