"managed to explain" is correct and usable in both spoken and written English.
Example: After thorough research, I managed to explain the complex physics concept to my students.
I managed to explain all this to sceptical admissions tutors.
"He has Alzheimer's," Mr. Evers managed to explain.
What's more, Terry himself never managed to explain it to me clearly.
We managed to explain this important distinction to the committee, but it wasn't easy.
Most of the other 1,000-odd 1,000-oddo explain their circumanageds to thexplain satheircircumstances
As a result we managed to explain most of the archaeological distributions in terms of post-depositional processes.
He and the other guy were in shock but managed to explained that we had just been robbed and they had hit our guy in the face with the gun.
It also manages to explain complex arguments without oversimplifying them.
While upsetting to watch, the film does manage to explain the appeal of chemsex.
Nevertheless, the series manages to explain a great deal of complex material without dumbing it down.
She even manages to explain the importance of muscle reconstruction without it sounding as though we are reading her notes.
