1,

20x7 + 7x - 6

= 20x7 - 8x + 15x - 6

= (20x7 - 8x )+ (15x - 6)

= 4x( 5x - 2 ) + 3( 5x - 2)

= ( 5x - 2 )( 4x + 3 )

2/

x4 - 5x2y2 + 4y4

= x4 - 4x2y2 - x2y2 + 4y2

= (x4 - 4x2y2 + 4y2) - x2y2

= ( x2 - 2y2)2 - x2y2

= ( x2 - 2y2)2 - (xy)2

= ( x2 - 2y2 - xy)( x2 - 2y2 + xy)

3/

x8 + y4 + 1

= ( x8 - x2 ) + ( x4 + 1 + x2 )

= x2( x6 - 1) + ( x4 + 1 + x2 )

= x2( x3 - 1 )( x3 + 1) + ( x4 + 1 + x2 )

= x2 . ( x - 1 ).( x2 + x + 1 ). ( x - 1).(x2 - x + 1) + ( x4 + 1 + x2 )

= x2 . \(\left[\left(x-1\right)\left(x+1\right)\right].\)\(\left[\left(x^2+x+1\right)\left(x^2-x+1\right)\right]\)+ ( x4 + 1 + x2 )

= x2(x2 - 1 )\(\left[\left(x^2+1\right)^2-x^2\right]\)+ ( x4 + 1 + x2 )

= x2(x2 - 1 )( x4 + 2x2 + 1 - x2 ) + ( x4 + 1 + x2 )

= x2(x2 - 1 )( x4 +x2 + 1 ) + ( x4 + 1 + x2 )

= ( x4 - x2 )( x4 +x2 + 1 ) + ( x4 + 1 + x2 )

= ( x4 +x2 + 1 )( x4 - x2 + 1 )