Question

What is the following product? (1√2+√6)(√6- √10)

Answer 1

Using the distributive property, this is
  (√2)×(√6 -√10) +(√6)(√6 -√10)
  = 2√3 - 2√5 +6 -2√15

The product is ...
  6 +2√3 -2√5 -2√15

_____
When simplifying the product, it helps to realize that any factor of 4 inside the radical can be brought outside as a factor of √4 = 2.
  (√2)(√6) = √12 = √(4*3) = 2√3
  (√2)(√10) = √20 = √(4*5) = 2√5
  (√6)(√10) = √60 = √(4*15) = 2√15

Answer 2

To multiply these, we use the FOIL method (first, outside, inside, last).  Also, 1√2 is equivalent to √2.  It's just like saying 1x is the same x.

Multiplying the first values will give you:

√2 * √6  =  √12  =  √4*3  =  √2*2*3  =  2√3

Multiplying the outside values will give you:

√2 * -√10  =  -√20  =  -√4*5  =  -√2*2*5  =  -2√5

Multiplying the inside values will give you:

√6 * √6  =  √36  =  6

Multiplying the last values will give you:

√6 * -√10  =  -√60  =  -√4*15  =  -2√15

Now you can combine all of the resulting values:

2√3 - 2√5 - 2√15 + 6

I believe this is the answer you're looking for.