Hi 130zewe, welcome to the forums.
I use single quotes for strings and don't embed variables ... it's much easier on the old eyeballs when trying to debug 😉
Also good for HTML. And you don't necessarily need to use $form .= either.
You could just do ...
<?php
$form = '
<b>Please enter in all the new user information...</b><br><br>
<form action="'.$_SERVER['PHP_SELF'].'" method="post"><br><br>
User Name: <input type="text" name="user_name" value="'.$user_name.'">
<input type="submit" value="Submit">
</form>
';
?>
It's all personal preference, I suppose, (lots of articles/tutorials seem to do it your way) but I think the above's much better for a lot of reasons.
Also, $PHP_SELF relies on the host's configuration and is deprecated (not to be relied upon in the future) so you should always use $_SERVER['PHP_SELF'] instead, which does the same job.
Paul 🙂
Edited for typo.