  In this paper we investigate a family of Moishezon twistor spaces on the
connected sum of 4 complex projective planes, which can be regarded as a direct
generalization of the twistor spaces on 3CP^2 of double solid type studied by
Poon and Kreussler-Kurke. These twistor spaces have a natural structure of
double covering over a scroll of 2-planes over a conic. We determine the
defining equations of the branch divisors in an explicit form, which are very
similar to the case of 3CP^2. Using these explicit description we compute the
dimension of the moduli spaces of these twistor spaces. Also we observe that
similarly to the case of 3CP^2, these twistor spaces can also be considered as
generic Moishezon twistor spaces on 4CP^2. We obtain these results by analyzing
the anticanonical map of the twistor spaces in detail, which enables us to give
an explicit construction of the twistor spaces, up to small resolutions.
