$$
\operatorname{Cov}(U, V)=\operatorname{Cov}(3 X+2 Y, X-4 Y)
$$
Use the linearity of covariance:
$$
\operatorname{Cov}(a X+b Y, c X+d Y)=a c \operatorname{Var}(X)+b d \operatorname{Var}(Y)+(a d+b c) \operatorname{Cov}(X, Y)
$$
Here:
$$
a=3, b=2, c=1, d=-4
$$
So:
$$
\operatorname{Cov}(U, V)=(3)(1) \operatorname{Var}(X)+(2)(-4) \operatorname{Var}(Y)+[(3)(-4)+(2)(1)] \operatorname{Cov}(X, Y)
$$
Simplify coefficients:
$$
\begin{gathered}
=3 \operatorname{Var}(X)-8 \operatorname{Var}(Y)+(-12+2) \operatorname{Cov}(X, Y) \\\\
=3 \operatorname{Var}(X)-8 \operatorname{Var}(Y)-10 \operatorname{Cov}(X, Y)
\end{gathered}
$$
$$
\rho=\frac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X) \operatorname{Var}(Y)}}
$$
$$
\operatorname{Cov}(X, Y)=\rho \sqrt{\operatorname{Var}(X) \operatorname{Var}(Y)}=0.5 \times \sqrt{4 \times 9}=0.5 \times 6=3
$$
$$
\begin{gathered}
\operatorname{Cov}(U, V)=3(4)-8(9)-10(3) \\\\
=12-72-30=-90
\end{gathered}
$$
