A schematic of the solution is shown below. The information
from the question is given in **black**. Deductions directly from this
information are given in **red**. Points that provide potential key information
are shown in **green** which leads to the structures which are linked via
the **blue** arrows to show the path required to work them all out. There
are, of course, other possible thought pathways.

**C**, **D** and **E **contain chirality centers.

The key step into this problem seems to be the Aldol reaction
by spotting the enone and reversing the intramolecular aldol to get **G**
and then** B** from which you can also get **F**. **B** also provides
access to **A** and **C** then **D** and **E**.

Alternatively, but a little more complex, once you get that
**A** is a conjugated diene, C5, then there are only two possible structures,
but one of those would give a Diels-Alder product as a racemic mixture...
but **B** doesn't have chirality, hence giving the structure of **A**.