Accurately predicting the state-of-health (SOH) and remaining useful life (RUL) of lithium-ion batteries is crucial for ensuring the safe and efficient operation of electric vehicles while minimizing associated risks. However, current deep learning methods are limited in their ability to selectively extract features and model time dependencies for these two parameters. Moreover, most existing methods rely on traditional recurrent neural networks, which have inherent shortcomings in long-term time-series modeling. To address these issues, this paper proposes a multi-task targeted learning framework for SOH and RUL prediction, which integrates multiple neural networks, including a multi-scale feature extraction module, an improved extended LSTM, and a dual-stream attention module. First, a feature extraction module with multi-scale CNNs is designed to capture detailed local battery decline patterns. Secondly, an improved extended LSTM network is employed to enhance the model's ability to retain long-term temporal information, thus improving temporal relationship modeling. Building on this, the dual-stream attention module-comprising polarized attention and sparse attention to selectively focus on key information relevant to SOH and RUL, respectively, by assigning higher weights to important features. Finally, a many-to-two mapping is achieved through the dual-task layer. To optimize the model's performance and reduce the need for manual hyperparameter tuning, the Hyperopt optimization algorithm is used. Extensive comparative experiments on battery aging datasets demonstrate that the proposed method reduces the average RMSE for SOH and RUL predictions by 111.3\% and 33.0\%, respectively, compared to traditional and state-of-the-art methods.