vectors

APQC is a quadrilateral and theyve just listed how its area is the sum of the two triangles, where the quadrilateral is split down a diagonal.
Presuming your question is a typo as the repeated side is not a diagonal, so area relationship is not correct.

So is the question wrong or am I wrong?

Just draw a sketch of a quadrilateral. You can obviously split it down either diagonal and the sum of the two triangles equals the original quadrilateral (so worked solution is correct). The repeated side is the diagonal. Your
area of APQ + area of PCQ
creates two triangles where PQ is the repeated side. PQ is a side of the quadrilateral, not a diagonal, so in general the sum of the two areas won't equal the quadrilateral area.

My apologies, I messed it the notation, thanks