Dense regions in networks are an indicator of interesting and unusual information. However, most existing methods only consider simple, undirected, unweighted networks. Complex networks in the real-world often have rich information though: edges are asymmetrical and nodes/edges have categorical and numerical attributes. Finding dense subgraphs in such networks in accordance with this rich information is an important problem with many applications. Furthermore, most existing algorithms ignore the higher-order relationships (i.e., motifs) among the nodes. Motifs are shown to be helpful for dense subgraph discovery but their wide spectrum in heterogeneous networks makes it challenging to utilize them effectively. In this work, we propose quark decomposition framework to locate dense subgraphs that are rich with a given motif. We focus on networks with directed edges and categorical attributes on nodes/edges. For a given motif, our framework builds subgraphs, called quarks, in varying quality and with hierarchical relations. Our framework is versatile, efficient, and extendible. We discuss the limitations and practical instantiations of our framework as well as the role confusion problem that needs to be considered in directed networks. We give an extensive evaluation of our framework in directed, signed-directed, and node-labeled networks. We consider various motifs and evaluate the quark decomposition using several real-world networks. Results show that quark decomposition performs better than the state-of-the-art techniques. Our framework is also practical and scalable to networks with up to 101M edges.