\(\text{a) }x^5+x^3-x^2-1\\ \\=\left(x^5-x^2\right)+\left(x^3-1\right)\\ \\=x^2\left(x^3-1\right)+\left(x^3-1\right)\\ \\=\left(x^2+1\right)\left(x^3-1\right)\\ \\=\left(x^2+1\right)\left(x-1\right)\left(x^2+x+1\right)\\ \)
\(\text{b) }x^2-x-12\\ \\=x^2-4x+3x-12\\ \\ =\left(x^2-4x\right)+\left(3x-12\right)\\ =x\left(x-4\right)+3\left(x-4\right)\\ \\=\left(x-4\right)\left(x+3\right)\\ \)
a) x5 + x3 - x2 - 1 = ( x5 + x3 ) - ( x2 + 1)
= x3 . ( x2 + 1 ) - ( x2 + 1 )
= ( x2 + 1 ) . ( x3 - 1 )
= ( x2 + 1 ) . ( x - 1 ) . ( x2 + x + 1 )
b) x2 - x - 12 = x2 - 4x + 3x - 12
= x . ( x - 4 ) + 3 . ( x - 4 )
= ( x - 4 ) . ( x + 3 )
c) 4x4 + 4x2y2 - 8y4 = 4x4 - 4x2y2 + 8x2y2 - 8y4
= 4x2 . ( x2 - y2 ) + 8y2 . ( x2 - y2 )
= ( x2 - y2 ) . ( 4x2 + 8y2 )
= 4 . ( x - y ) . ( x + y ) . ( x2 + 2y2 )
