  We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg
structure, a notion introduced in [GNPR10] leading to a good calculation of the
homotopy category in terms of (co)fibrant objects. This result provides a
conceptual framework from which Beilinson's [Bei86] and Carlson's [Car80]
results on mixed Hodge complexes and extensions of mixed Hodge structures
follow easily.
