There are many things that solve will not do automatically because they
are heuristics which may or may not introduce extraneous roots, and/or
because they may be expensive. Or they just weren't programmed (yet).
This does not prevent you from trying them yourself before calling
solve, of course.
If you want to poke at the solve program, there are also issues like
this one.
(-sqrt(2)*sqrt(25*sqrt(2)*sqrt(5)-3)-6*sqrt(5)*sqrt(sqrt(2)*sqrt(5)+3)+13*sqrt(2)*sqrt(sqrt(2)*sqrt(5)+3))*x=4
solve(%,x);
Solve gives you one solution, but there are no solutions, because the
equation is equivalent to 0*x=4.
(Try proving that :)
RJF
(who originally wrote the solve program in about 1969)