I can find $\mu_1, \mu_2, \operatorname{Var}(Y_1) \ \text{and } \operatorname{Var}(Y_2)$ but I am not sure how to get the co-variance of $Y_1$ and $Y_2$ in order to find the correlation of $Y_1$ and $Y_2$. So far I have
$\operatorname{Cov}(Y_1,Y_2)=E(Y_1,Y,2)-E(Y_1)E(Y_2)$
After plugging in the $\mu$s and Vars and got $E((X1)^2)$. But I have no clue how to continue with the calculation. Am I on the right track?
The problem term is $$ E\left[Y_{1}Y_{2}\right]=E\left[\left(X_{1}+2X_{3}\right)\left(X_{1}-2X_{3}\right)\right]=E\left[X_{1}^{2}\right]-4E\left[X_{3}^{2}\right]. $$ So, what is $E\left[X_{k}^{2}\right]$? Recall $$ \sigma^{2}=\text{Var}\left(X_{k}\right)=E\left[X_{k}^{2}\right]-E\left[X_{k}\right]^{2}=E\left[X_{k}^{2}\right]-\mu^{2} $$ so that $$ E\left[X_{k}^{2}\right]=\sigma^{2}+\mu^{2}. $$