Show advanced search form outside fieldset When I search for a keyword in core Drupal search, it shows a results page with a link named Advanced search that hides a fieldset with more form options. I want it to be shown in the basic search form, not only on search result page, and it should be expanded by default.

Note: I want all this with core search in Drupal 7.
take a look at the Advanced Search Module:

This module alters the default search box in many ways. If you need to
have options available like in advanced search, but directly in the
search box, this module is for you.
Basic options The module adds
options to select:
which content type(s) to search, which specific module search to use
(node, help, user or any module that implements search), which input
type to use (with Elements), advanced criteria For all these choices,
there are options to switch between a select box, checkboxes or radio
buttons, and also customize the selector label and the default - Any -
text.
Advanced options There are also options to:
change the default search box label, add a default text in the search
box, add advanced search criteria, change the default submit button
text, use an image instead of the submit button, change the order of
all the elements, include some elements in a popup block, add a filter
to the results page, show/hide basic and/or advanced search in the
results page, show/hide meta data in the results page, multiple search
paths