The question defines a series of random variables U, V, X, Y, and Z. X, Y, and Z are "uncorrelated". U = X + Z, V = Y + Z. The variances of X, Y, and Z are known (let's say for simplicity that they are 1, 2, and 3, respectively). The question is asking for the covariance of (U,V).
I'm aware that I can compute cov(U,V) as cov(X+Z,Y+Z) = cov(X,Y) + cov(X,Z) + cov(Y,Z) + cov(Z,Z) and that cov(Z,Z) = var(Z) = 3. But I'm not aware of anything else I can be doing to work out these other covariances. I understand that covariance depends on expected values, and I have no information regarding any of the data, just given the variances themselves. Where do I go from here?
Since we know in advance that $X$, $Y$ and $Z$ are uncorrelated, we have $$\operatorname{cov}(U, V)$$ $$=\operatorname{cov}(X+Z, Y+Z)$$ $$=\operatorname{cov}(X, Y)+\operatorname{cov}(X, Z)$$ $$+\operatorname{cov}(Z, Y)+\operatorname{cov}(Z, Z)$$ $$=\operatorname{var}(Z)$$