ok – you want to have the post’s image in the search result?
(it seems that you were not referring to the ‘featured image’ aka post_thumbnail, but to a thumbnail in the post)
before you proceed, strongly consider to create a child theme for your customisations; https://codex.www.ads-software.com/Child_Themes
it is not recommended to edit any of the default themes directly.
then edit content.php (a copy of content.php in your child theme):
this section decides to show only excerpts for search results:
<?php if ( is_search() ) : // Only display Excerpts for Search ?>
<div class="entry-summary">
<?php the_excerpt(); ?>
</div><!-- .entry-summary -->
<?php else : ?>
<div class="entry-content">
<?php the_content( __( 'Continue reading <span class="meta-nav">→</span>', 'twentytwelve' ) ); ?>
<?php wp_link_pages( array( 'before' => '<div class="page-links">' . __( 'Pages:', 'twentytwelve' ), 'after' => '</div>' ) ); ?>
</div><!-- .entry-content -->
<?php endif; ?>
remove the conditional structure and leave only:
<div class="entry-content">
<?php the_content( __( 'Continue reading <span class="meta-nav">→</span>', 'twentytwelve' ) ); ?>
<?php wp_link_pages( array( 'before' => '<div class="page-links">' . __( 'Pages:', 'twentytwelve' ), 'after' => '</div>' ) ); ?>
</div><!-- .entry-content -->