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\(4x^4+4x^3+5x^2+8x-6\)
\(=4x^4-2x^3+6x^3-3x^2+8x^2-4x+12x-6\)
\(=2x^3\left(2x-1\right)+3x^2\left(2x-1\right)+4x\left(2x-1\right)+6\left(2x-1\right)\)
\(=\left(2x^3+3x^2+4x+6\right)\left(2x-1\right)\)
\(=\left[x^2\left(2x+3\right)+2\left(2x+3\right)\right]\left(2x-1\right)\)
\(=\left(x^2+2\right)\left(2x+3\right)\left(2x-1\right)\)
\(4x^4+6x^3-4x^2+9x-15\)
\(=4x^4-4x^3+10x^3-10x^2+6x^2-6x+15x-15\)
\(=4x^3\left(x-1\right)+10x^2\left(x-1\right)+6x\left(x-1\right)+15\left(x-1\right)\)
\(=\left(4x^3+10x^2+6x+15\right)\left(x-1\right)\)
\(=\left[2x^2\left(2x+5\right)+3\left(2x+5\right)\right]\left(x-1\right)\)
\(=\left(2x^2+3\right)\left(2x+5\right)\left(x-1\right)\)
a) \(7x^3y-3xyz-21x^2+9z\)
\(=7x^2\left(xy-3\right)-3z\left(xy-3\right)\)
\(=\left(7x^2-3z\right)\left(xy-3\right)\)
b) \(4x^2-2x-y^2-y\)
\(=\left[\left(2x\right)^2-y^2\right]-\left(2x+y\right)\)
\(=\left(2x-y\right)\left(2x+y\right)-\left(2x+y\right)\)
\(=\left(2x+y\right)\left(2x-y-1\right)\)
c) \(9x^2-25y^2-6x+10y\)
\(=\left(3x\right)^2-\left(5y\right)^2-2\left(3x-5y\right)\)
\(=\left(3x-5y\right)\left(3x+5y\right)-2\left(3x-5y\right)\)
\(=\left(3x-5y\right)\left(3x+5y-2\right)\)
d) \(\left(5x-4\right)^2+\left(16-25x^2\right)+\left(5x-4\right)\left(3x+2\right)\)
\(=\left(5x-4\right)\left[\left(5x-4\right)+\left(3x+2\right)\right]+\left(4^2-\left(5x\right)^2\right)\)
\(=\left(5x-4\right)\left(8x-2\right)+\left(4-5x\right)\left(4+5x\right)\)
\(=\left(4-5x\right)\left(2-8x\right)+\left(4-5x\right)\left(4+5x\right)\)
\(=\left(4-5x\right)\left[\left(2-8x\right)+\left(4+5x\right)\right]\)
\(=\left(4-5x\right)\left(6-3x\right)\)
\(x^4+1\)
\(=x^4+2x^2+1-2x^2\)
\(=\left(x^2+1\right)^2-2x^2\)
\(=\left(x^2+1\right)^2-\left(\sqrt{2}x\right)^2\)
\(=\left(x^2+1-\sqrt{2}x\right)\left(x^2+1+\sqrt{2}x\right)\)