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a) \(4x^4-21x^2y^2+y^4\)
\(=\left(2x^2\right)^2-2\cdot2x^2\cdot y^2+y^2-25x^2y^2\)
\(=\left(2x^2-y^2\right)^2-\left(5xy\right)^2\)
\(=\left(2x^2-5xy-y^2\right)\left(2x^2+5xy-y^2\right)\)
b) \(x^5-5x^3+4x\)
\(=x^5-4x^3-x^3+4x\)
\(=x^3\left(x^2-4\right)-x\left(x^2-4\right)\)
\(=\left(x^2-4\right)\left(x^3-x\right)\)
\(=x\left(x-2\right)\left(x+2\right)\left(x^2-1\right)\)
\(=x\left(x-2\right)\left(x+2\right)\left(x-1\right)\left(x+1\right)\)
\(x^3-x^2-21x+45\)
\(=\left(x^3-3x^2\right)+\left(2x^2-6x\right)+\left(-15x+45\right)\)
\(=x^2\left(x-3\right)+2x\left(x-3\right)-15\left(x-3\right)\)
\(=\left(x^2+2x-15\right)\left(x-3\right)\)
\(=\left[\left(x^2-3x\right)+\left(5x-15\right)\right]\left(x-3\right)\)
\(=\left[x\left(x-3\right)+5\left(x-3\right)\right]\left(x-3\right)\)
\(=\left(x+5\right)\left(x-3\right)^2\)
4x^4+y^4= 4x^4 +4x^2y^2+y^4-4x^2y^2= ( 2x^2 + y^2 ) ^2 - ( 2xy ) ^2
= (2x^2 + 2xy +y^2)( 2x^2 - 2xy + y^2)
`x^3+y^3+21x+21y`
`=(x+y)(x^2-xy+y^2)+21(x+y)`
`=(x+y)(x^2-xy+y^2+21)`
\(x^3+y^3+21x+21y\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)+21\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2+21\right)\)