We propose a method to image a complex scene with spotlight synthetic aperture radar (SAR) despite the presence of multiple moving targets. Many recent methods use sparsity-based reconstruction coupled with phase error corrections of moving targets to reconstruct stationary scenes. However, these methods rely on the assumption that the scene itself is sparse and thus unfortunately cannot handle realistic SAR scenarios with complex backgrounds consisting of more than just a few point targets. Our method makes use of sparse and low-rank (SLR) matrix decomposition, an efficient method for decomposing a low-rank matrix and sparse matrix from their sum. For detecting the moving targets and reconstructing the stationary background, SLR uses a convex optimization model that penalizes the nuclear norm of the low rank background structure and the L1 norm of the sparse moving targets. We propose an L1-norm regularization reconstruction method to form the input data matrix, which is grossly corrupted by the moving targets. Each column of the input matrix is a reconstructed SAR image with measurements from a small number of azimuth angles. The use of the L1-norm regularization and a sparse transform permits us to reconstruct the scene with significantly fewer measurements so that moving targets are approximately stationary. We demonstrate our SLR-based approach using simulations adapted from the GOTCHA Volumetric SAR data set. These simulations show that SLR can accurately image multiple moving targets with different individual motions in complex scenes where methods that assume a sparse scene would fail.