SOLUTION: My 8th grade son has the following problem: -5(x + 2) = -5x - 10 He is telling me his teacher said whatever the x is does not get added to the 2 because there is no x beside

Sure.  We cannot add x + 2,  but we could have added x + 2x 
if there had been an x next to the 2.

We could add x+2x and get 3x because x means 1x, and so x+2x is 1x+2x,
which is like adding 1 apple + 2 apples and getting 3 apples.

However x is an UNKNOWN number and 2 is a KNOWN number.  We cannot 
add a KNOWN number to an UNKNOWN number and get a KNOWN number, which 
would be like "adding apples to oranges and getting bananas" -- which is
bananas! :) So all we can do is write what is to be done, namely as a
"formula" x + 2 and leave it that way until maybe later (in some cases) 
we find out what x is. 

Now, what your son's teacher is explaining is the DISTRIBUTIVE PRINCIPLE. 

As above we cannot add the 2 to the x because we don't know what the x 
represents so we can't add it to the 2.  So we have to leave it unadded,
as a "formula" x+2.  

You would be right if we knew what number x represented.  For instance, if
we had known, say, that x represents 7, then we would have

-5(7 + 2)

and then we would do as you said and add what x is 
(in this case 7) to the 2 and get -5(9) and then we'd have -45.
If x were some other number besides 7 we would get a different answer.

Since we do not know what x represents. we would normally have to 
leave it as this "formula", and couldn't even multiply by the -5.
We would have to leave it just like this:

-5(x + 2)

However, the DISTRIBUTIVE PRINCIPLE allows us to go ahead and multiply 
WITHOUT adding first! 

We multiply the -5 by the x and get -5x.  Then we multiply the -5 by the +2
and get -10.  So we have

-5(x + 2) = -5x - 10.

The distributive principle is a way to go ahead and multiply each term
by the -5 even though we cannot add the "apples" x to the "oranges" 2.

If you have any more questions, you can ask me in the thank-you note.

Edwin