In Exercises 4568 perform the indicated operations and expre

In Exercises 45-68, perform the indicated operations and express results in simplest form.

52) ( 6x^2 - xy - y^2)/(2x^2 +xy - y^2) / (16y^2 - 4x^2)/(2x^2 + 6xy +4y^2)

factorise the expressions:

-----6x^2 - xy - y^2 = 6x^2 -3xy +2xy - y^2

= 3x(2x - y) +y(2x - y) = (3x +y)(2x -y)

----2x^2 +xy - y^2 = 2x^2 +2xy -xy -y^2 = (2x(x +y) -y(x+y)

= (2x -y)(x+y)

----(16y^2 - 4x^2) = (4y -2x)( 4y +2x)

----- 2x^2 + 6xy +4y^2 = 2x^2 + 4xy +2xy + 4y^2

= 2x(x+2y)+2y(x+2y)

= (2x +2y)(x +2y)

So, ( 6x^2 - xy - y^2)/(2x^2 +xy - y^2) / (16y^2 - 4x^2)/(2x^2 + 6xy +4y^2)

= (3x +y)(2x -y)*(2x +2y)(x +2y) /(2x -y)(x+y)(4y -2x)( 4y +2x)

= 2(3x +y)(x+2y)/(4y -2x)(4y +2x)

=2(3x +y)(x+2y)/(2(2y -x)2(2y+x)

= (3x+y)/2(2y -x)