Multiply.

Thanks Aziz,

I do these a different way.  My way is perhaps more versatile because you can use it no matter how many terms are inside the brackets.

I take the 1st term in the first bracket and multiply it by the second bracket then

I take the 2nd term in the first bracket and mltiply it by the second bracket then

I can keep doing this until I run out of terms in the first bracket.

Like this

$$\textcolor[rgb]{0,0,1}{(2+4\sqrt{10})}(4\sqrt5 -2)\\\\
=\textcolor[rgb]{0,0,1}{2}(4\sqrt5-2)\textcolor[rgb]{0,0,1}{+4\sqrt{10}}(4\sqrt5-2)\\\\
=8\sqrt5-4+16\sqrt{50}-8\sqrt{10}$$

Perform the FOIL method:

First - 2*4sqrt(5) = 8sqrt(5)

Outside - 2*(-2) = -4

Inside - 4sqrt(10)*4sqrt(5) = 16sqrt(50)

Last - 4sqrt(10)*(-2) = -8sqrt(10)

Adding all the terms: 8sqrt(5) - 4 + 16sqrt(50) - 8sqrt(10)

16sqrt(50) can be converted to: 16*sqrt(25*2) = 16*sqrt(25)*sqrt(2); we can take out the 25 from the square root as a 5 since sqrt(25) = 5 --> 16*5sqrt(2) = 80sqrt(2)

Now, we have 8sqrt(5) - 4 + 80sqrt(2) - 8sqrt(10)

8[sqrt(5) - sqrt(10) + 10sqrt(2) - (1/2)]*

*I have just done the distributive property backwards so that when you distribute it, you get the same result: 8sqrt(5) - 4 + 80sqrt(2) - 8sqrt(10)

For 80sqrt(2), since 8 is a common term that can be taken out, we can make it 8(10sqrt(2)).

For -4, we know that 8*(-1/2) = -4; therefore, we can make it 8(-1/2).

Now, combining like terms for the 8sqrt(5) and - 8 sqrt(10) is easy since both have an 8. 

That is how we end up with 8[sqrt(5) - sqrt(10) + 10sqrt(2) - (1/2)] as the final answer.