Verified answer

See work and solution below

  the question should NEVER be -     why is the book answer correct?

it SHOULD be - why is MY answer wrong?

 THAT way you find out the mistake in YOUR thinking

 so where is YOUR answer?

(telling you how the BOOK worked it out tells you NOTHING about how you got it wrong and how to correct YOUR mistake)

35.

....x^2 - 4............. 2 - x

 --------------- ÷  -------------

........12................2x + 4

....(x^2 - 4).........2x + 4

=-------------- * -------------

..........12............2 - x

....(x + 2)(x - 2)........2(x + 2)

=-------------------- * ------------- cancel out x - 2

...........12...............- (x - 2)

.....(x + 2)^2

=- -------------- answer//

...........6

35. [(x^2-4)/(12)]/[(2-x)/(2x+4)] = (x-2)(x+2)2(x+2)/[12(2-x) = - (1/6)(x+2)^2. where

x =/= (+/-)2. 

( x² - 4 )  / 12

-----------------

( 2- x ) / ( 2x  + 4 )

  ( x. - 2 ) ( x  + 2 ) 2 ( x. + 2 )

-------------------------------------

          12 ( 2  - x )

 -  (x  + 2 ) ²

----------------

6

Ans is given below 

You're dividing by (2-x)/(2x+4).

And, since you cannot divide by zero,

as you found, x cannot be 2.

But, for  x =  -2, (2-x)/(2x+4) becomes  4/0

and you cannot divide by that either.

So, x  ≠  ±2

(x^2 - 4)/12 divided by (2-x)/(2x+4)

= (x^2 - 4)/12 times (2x+4)/(2-x)

= (x+2)(x-2)/12 times (-2)(x+2)/(x-2)

Cancel the (x-2), you have

(-2)(x+2)^2 / 12,

then divide top and bottom by 2, and you have

-(x+2)^2 / 6.

The reason why you can't have x=2 or x= -2 is that either of these would put a zero in a denominator somewhere along the way.

[(x^2 - 4) / 12] / [(2 - x) / (2x + 4)]  Original

[(x^2 - 4) / 12] [(2x + 4) / (2 - x)]  Invert and Multiply 

[(x + 2)(x - 2) / 12]  [-2(x + 2) / (x - 2)]  Factoring and simplifying

-(x +2)^2 / 6  Cancel (x - 2) in numerator and denominator as well as further simplification