Random variables and co-variance, Statistics 318

For the given example in the book

John E. Freund's Mathematical Statistics with Applications, 8th edition, by Miller and Miller. ISBN: 9780321807090

I've highlighted using colors what numbers corespond with what is given (Blue, Green, and Yellow).

What I don't understand is where the highlighted red numbers come from.

The example states it uses "Theorem 15" I'm just confused on how.

"The following is another important theorem about linear combinations of random variables; it concerns the covariance of two linear combinations of n random variables."

• You´ve comprehended the answer of parsaid within 5 seconds. Congrats. – callculus Oct 21 '18 at 22:30
• @callculus well to be fair I continued to work through the problem and I figured it out on my own and as I was working through it I saw the answer was posted, and the answer goes along with what I figured out so I figured I would select it as answered to help others out and since it is correct. So not quite 5 seconds haha but thanks! – CTOverton Oct 21 '18 at 22:38
• I was just wondering. Thanks for your reply. – callculus Oct 21 '18 at 22:40

Hint: Define $$X_{1}=X$$, $$X_{2}=Y$$, and $$X_{3}=Z$$. Also define $$a_{1}=1$$, $$a_{2}=4$$, $$a_{3}=2$$, $$b_{1}=3$$, $$b_{2}=-1$$, and $$b_{3}=-1$$. Then, $$\text{cov}(X+4Y+2Z,3X-Y-Z) =\text{cov}(\underbrace{a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}}_{Y_1},\underbrace{b_{1}X_{1}+b_{2}X_{2}+b_{3}X_{3}}_{Y_2}).$$ The above now looks like the expression in Theorem 15. Can you finish?