<?php
generate_random_word();
$prefix = mt_rand();
$captcha_instance->generate_image( $prefix, $word );
?>

On your contact form page.

Then on the results page (where you do your checks and then collect the data)
<?php
check( $prefix, $the_answer_from_respondent );

if ($correct="true")
{
<run code for correct response >
}
else
  {
<run code for negative response >
  }
?>