a) 9 . ( 2x + 3 )2 - 4 . ( x + 1 )2
= 32 . ( 2x + 3 )2 - 22 . ( x + 1 )
= [ 3 . ( 2x + 3 ) ] 2 - [ 2 . ( x + 1 ) ]
= ( 6x + 9 )2 - ( 2x + 2 )2
= [ ( 6x + 9 ) - ( 2x + 2 ) ] . [ ( 6x + 9 ) + ( 2x + 2 ) ]
= ( 6x + 9 - 2x - 2 ) . ( 6x + 9 + 2x + 2 )
= ( 4x + 7 ) . ( 8x + 11 )
b) 27x3 + \(\dfrac{y^3}{8}\) = ( 3x )3 + \(\left(\dfrac{y}{2}\right)^{^{ }3}\)
= ( 3x + \(\dfrac{y}{2}\) ) . ( 9x2 - \(\dfrac{3}{2}\)xy + \(\dfrac{y^2}{4}\) )
c) ( x + y )3 - ( x - y )3
= [ ( x + y ) - ( x - y )] . [ ( x + y )2 + ( x + y ) . ( x - y ) + ( x - y )2
= ( x + y - x + y ) . [ ( x2 + 2xy + y2 ) ] + ( x2 - y2 ) + ( x2 - 2xy + y2 ) ]
= 2y . ( x2 + 2xy + y2 + x2 - y2 + x2 - 2xy + y2 )
= 2y . ( 3x2 + y2 )
= 6y . ( x2 + y2 )
