The requested php code:
   $post_data = "secret=__secret-key__&response=".$_POST['g-recaptcha-response']."&remoteip=".$_SERVER['REMOTE_ADDR'] ;
    $ch = curl_init();  
    curl_setopt($ch, CURLOPT_URL, "
https://www.google.com/recaptcha/api/siteverify");
    curl_setopt($ch, CURLOPT_POST, true);
    curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
    curl_setopt($ch, CURLOPT_HTTPHEADER, array('Content-Type: application/x-www-form-urlencoded; charset=utf-8', 'Content-Length: ' . strlen($post_data)));
    curl_setopt($ch, CURLOPT_POSTFIELDS, $post_data); 
    $googresp = curl_exec($ch);       
    $decgoogresp = json_decode($googresp);
    curl_close($ch);
if ($decgoogresp->success == true)
  {
  // Captcha accepted
  }
else
  {
  // Let the user know ...
  }