To answer the question you asked:
Do look for the painless approach only if you feel a pain. Do you? Express it then, to make others understand what certainly makes you uneasy, to let them offer certain solution for this very issue. If not - just leave it as is. Exactly the same goes for the efficiency. And easiness too. Altering your code without a reason, just out of a whim, out of nowhere, may cause much more troubles than you imagine with your current code.
To answer the question you didn't ask:
Your approach, as well as in all the answers, suffers from one essential flaw: you don't format your query properly. Which may lead to unpredictable consequences. To format your query properly you have to use prepared statements
$stm = $DBconnect->prepare("insert into solution_tags values(?)");
$stm->bind_param('s', $tag);
foreach($array as $tag){
$stm->execute();
}
again: this is not a matter of "efficiency" or "ease" (as a matter of fact, on a more or less complex query this approach will be a pain compared to your current one), but matter of essential approach you ought to follow with every your query.
On a side note, you may wish to add some field(s) to this table to link these tags to some other entities.