Expectation of quadratic form of correlated variables

Question

Suppose $U$ and $V$ are two $n\times 1$ dependent vectors, in the sense that $E\left( UV^{\prime}\right) \neq \mathbf{0}.$ For a given constant $n\times n$ matrix $A,$ is there is any simple way to calculate $E\left[ \left( U^{\prime }AV\right) ^{2}\right] ,$ or even to find a upper bound and lower bound for this quadratic expectation? Thanks a lot for your attention.