Factoring out a Cohen forcing

Question

Suppose that in a forcing extension $V[G]$ by some ccc forcing $P$ there is a Cohen real over $V$. By a general argument we can factor $P$ into an iteration $P=\mathrm{Add}(\omega,1)*\dot{Q}$ for some ccc quotient forcing $\dot{Q}$. Can we do better and factor $P$ as a product $\mathrm{Add}(\omega,1)\times R$ for some forcing $R$? Is every extension containing a Cohen real a Cohen extension of some intermediate model?

Answer 1

Suppose you first add a Cohen real and then, over the resulting model, force Martin's axiom and not CH in the usual way (by a ccc, finite-support iteration). The resulting model has no Souslin trees, because MA$(\aleph_1)$ implies Souslin's hypothesis. So this model is not of the form "add a Cohen real to some model", because adding a Cohen real always produces a Souslin tree.