The problem of learning and forecasting underlying trends in time series data arises in a variety of applications, such as traffic management, energy optimization, etc. In literature, a trend in time series is characterized by the slope and duration, and its prediction is then to forecast the two values of the subsequent trend given historical data of the time series. For this problem, existing approaches mainly deal with the case in univariate time series. However, in many real-world applications, there are multiple variables at play, and handling all of them at the same time is crucial for an accurate prediction. A natural way is to employ multi-task learning (MTL) techniques in which the trend learning of each time series is treated as a task. The key point of MTL is to learn task relatedness to achieve better parameter sharing, which however is challenging in trend prediction task. First, effectively modeling the complex temporal patterns in different tasks is hard as the temporal and spatial dimensions are entangled. Second, the relatedness among tasks may change over time. In this paper, we propose a neural network, DeepTrends, for multivariate time series trend prediction. The core module of DeepTrends is a tensorized LSTM with adaptive shared memory (TLASM). TLASM employs the tensorized LSTM to model the temporal patterns of long-term trend sequences in an MTL setting. With an adaptive shared memory, TLASM is able to learn the relatedness among tasks adaptively, based upon which it can dynamically vary degrees of parameter sharing among tasks. To further consider short-term patterns, DeepTrends utilizes a multi-task 1dCNN to learn the local time series features, and employs a task-specific sub-network to learn a mixture of long-term and short-term patterns for trend prediction. Extensive experiments on real datasets demonstrate the effectiveness of the proposed model.