how do i simplify this, showing all steps?

Question

4y-8 / y+2 divided by y-2 / y squared -2

I get that answer... but then when i check it in the back of the book it has the answer as: 4 (y-2)

Answer 1

4y-8 / y+2 divided by y-2 / y squared -2

I take this to be:
[ (4y - 8) / (y + 2) ] / [ (y - 2) / (y^2 - 2) ]
Invert the fraction in the second set of square brackets, and change the division between the sets of square brackets to multiply. That gives:
[ (4y - 8) / (y + 2) ] * [ (y^2 - 2) / (y - 2) ]
Now combine the numerators of the two fractions and combine their denominators, getting:
[ (4y - 8)(y^2 - 2) ] / [ (y + 2)(y - 2) ]
Now factorise where possible (there isn't much):
[ 4(y - 2)(y^2 - 2) ] / [ (y + 2)(y - 2) ]
You can now remove the y - 2 factor which is common to numerator and denominator, getting:
4(y^2 - 2) / (y + 2).

Answer 2

4(y-2)/ y+2 / y-2/y^2-2
4(y-2)/y+2 * y^2-2/y-2

the two y-2 cancel out

4(y^2-2)/y-2