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."
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?