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\(x^2-y^2+2x+1\)
\(=\left(x^2+2x+1\right)-y^2\)
\(=\left(x+1\right)^2-y^2\)
\(=\left(x+1-y\right)\left(x+1+y\right)\)
\(=\left(x+y-1\right)\left(x+y+1\right)\)
\(---\)
\(\left(x^2+9\right)^2-36x^2\)
\(=\left(x^2+9\right)^2-\left(6x\right)^2\)
\(=\left(x^2+9-6x\right)\left(x^2+9+6x\right)\)
\(=\left(x^2-2\cdot x\cdot3+3^2\right)\left(x^2+2\cdot x\cdot3+3^2\right)\)
\(=\left(x-3\right)^2\left(x+3\right)^2\)
\(---\)
\(x^3-4x^2+8x-8\)
\(=\left(x^3-8\right)-\left(4x^2-8x\right)\)
\(=\left(x^3-2^3\right)-4x\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2-x\cdot2+2^2\right)-4x\left(x-2\right)\)
\(=\left(x-2\right)\left[\left(x^2-2x+4\right)-4x\right]\)
\(=\left(x-2\right)\left(x^2-2x+4-4x\right)\)
\(=\left(x-2\right)\left(x^2-6x+4\right)\)
\(---\)
\(x^4-x^3-x+1\)
\(=\left(x^4-x^3\right)-\left(x-1\right)\)
\(=x^3\left(x-1\right)-\left(x-1\right)\)
\(=\left(x-1\right)\left(x^3-1\right)\)
\(=\left(x-1\right)\left(x-1\right)\left(x^2+x\cdot1+1^2\right)\)
\(=\left(x-1\right)^2\left(x^2+x+1\right)\)
\(---\)
\(2x^3+x^2-4x-12\)
\(=2x^3-4x^2+5x^2-10x+6x-12\)
\(=\left(2x^3-4x^2\right)+\left(5x^2-10x\right)+\left(6x-12\right)\)
\(=2x^2\left(x-2\right)+5x\left(x-2\right)+6\left(x-2\right)\)
\(=\left(x-2\right)\left(2x^2+5x+6\right)\)
\(---\)
\(25x^2\left(x-y\right)-x+y\)
\(=25x^2\left(x-y\right)-\left(x-y\right)\)
\(=\left(x-y\right)\left(25x^2-1\right)\)
\(=\left(x-y\right)\left[\left(5x\right)^2-1\right]\)
\(=\left(x-y\right)\left(5x-1\right)\left(5x+1\right)\)
\(---\)
\(16x^2\left(z^2-y^2\right)-z^2+y^2\)
\(=\left(4x\right)^2\cdot\left(z^2-y^2\right)-\left(z^2-y^2\right)\)
\(=\left(z^2-y^2\right)\left[\left(4x\right)^2-1\right]\)
\(=\left(z-y\right)\left(z+y\right)\left(4x-1\right)\left(4x+1\right)\)
\(---\)
\(x^3+x^3y-x^2z-xyz\)
\(=\left(x^3-x^2z\right)+\left(x^2y-xyz\right)\)
\(=x^2\left(x-z\right)+xy\left(x-z\right)\)
\(=\left(x-z\right)\left(x^2+xy\right)\)
\(=x\left(x-z\right)\left(x+y\right)\)
\(---\)
\(12x^5y+24x^4y^2+12x^3y^3\)
\(=12x^5y+12x^4y^2+12x^4y^2+12x^3y^3\)
\(=\left(12x^5y+12x^4y^2\right)+\left(12x^4y^2+12x^3y^3\right)\)
\(=12x^4y\left(x+y\right)+12x^3y^2\left(x+y\right)\)
\(=\left(12x^4y+12x^3y^2\right)\left(x+y\right)\)
\(=12x^3y\left(x+y\right)\left(x+y\right)\)
\(=12x^3y\left(x+y\right)^2\)
#\(Toru\)
