What is the following quotient? 2 - sqrt 8 / 4 + sqrt 12

C is the answer.

We multiply the top and bottom of this by the conjugate (switch the sign of the square root):

(2-sqrt8) (4-sqrt12) / (4+sqrt12) (4-sqrt12)

Multiying the top we have

8-2sqrt12-4sqrt8+sqrt (8*12)

Multiplying the bottom we have:

16-4sqrt12+4sqrt12-12

On top, we simplify the radicals and get

8-4sqrt3-8sqrt2+4sqrt6

On bottom, we have

16-12=4

Now we divide everything on top by 4 and get

2-sqrt3-2sqrt2+sqrt6