Help Integrating Re-Captcha Into My PHP Form

<div id="cos">
<center><u><h1>Please Fill In Your Information:</h1></u></center>
<div id="costext1">
<form method="POST" action="leads.php">
<table border="0" cellpadding="5">
<tr>
<td>
   First Name: <br /> <input type="text" name="firstname" size="19"></td>
<td>
  Last Name: <br /> <input type="text" name="lastname" size="19"></td>
<td>
Email: <br /> <input type="text" name="email" size="19">
</td>
<tr>
<td>
 Phone Number: <br />  <input type="text" name="phone" size="19">
</td>
<td>
 Age: <br /> <input type="text" name="age" size="19">
</td>
<td>Sex: <br /> <input type="radio" name="sex" value="male" /> Male
<input type="radio" name="sex" value="female" /> Female
</td>
</tr>
<tr>
<td>
 Address: <br /> <input type="text" name="address" size="19">
</td>
<td>
City: <br /> <input type="text" name="city" size="19">
</td>
</tr>
</table>
</div>
<div id="costext2">
<table cellpadding="5" border="0">
<tr>
<td>
 What procedure are you interested in?: <br />
&nbsp;&nbsp;<SELECT NAME="procedure">
<OPTION VALUE="Botox">Botox
<OPTION VALUE="Laser Hair Removal">Laser Hair Removal
<OPTION VALUE="Hyaluronic Acid">Hyaluronic Acid
<OPTION VALUE="Chemical Peel">Chemical Peel
<OPTION VALUE="Laser Skin Resurfacing">Laser Skin Resurfacing
<OPTION VALUE="Breast Augmentation">Breast Augmentation
<OPTION VALUE="Breast Reduction">Breast Reduction
<OPTION VALUE="Breast Lift">Breast Lift
<OPTION VALUE="Breast Nipple Elevation">Breast Nipple Elevation
<OPTION VALUE="Buttocks Augmentation">Buttocks Augmentation
<OPTION VALUE="Liposuction">Liposuction
<OPTION VALUE="Eyelid Surgery">Eyelid Surgery
<OPTION VALUE="Non Invasive Tightening">Non-Invasive Tightening
<OPTION VALUE="Nose Job">Nose Job
<OPTION VALUE="Abdominoplasty">Tummy Tuck (Abdominoplasty)
<OPTION VALUE="Microdermabrasion">Microdermabrasion
<OPTION VALUE="Gynecomastia">Gynecomastia
<OPTION VALUE="Hair Transplant">Hair Transplant
<OPTION VALUE="Arm Procedures">Arm Procedures
<OPTION VALUE="Ear Surgery">Ear Surgery
<OPTION VALUE="Eyebrow Lift">Eyebrow Lift
<OPTION VALUE="Light Facial">Liquid and Light Facial
<OPTION VALUE="S Lift">Boutique S-Lift
<OPTION VALUE="Chin Implants">Chin Implants
<OPTION VALUE="Cheek Implants">Cheek Implants
<OPTION VALUE="Circumcision">Circumcision
<OPTION VALUE="Calf Implants">Calf Implants
<OPTION VALUE="Enlargement">Enlargement
<OPTION VALUE="Face Lift">Face Lift
<OPTION VALUE="Neck Lift">Neck Lift
<OPTION VALUE="Female Parts Surgery">Female Parts Surgery
<OPTION VALUE="Other Implants">Implants
<OPTION VALUE="Keloids">Keloids
<OPTION VALUE="Lip Enhancement">Lip Enhancement
<OPTION VALUE="Liposculpture">Liposculpture
<OPTION VALUE="Lesions">Lesions
<OPTION VALUE="Nose Implants">Nose Implants
<OPTION VALUE="Nose Reshaping">Nose-Reshaping
<OPTION VALUE="Rhinoplasty">Nose-Rhinoplasty
<OPTION VALUE="Suture Lift">Suture Lift
<OPTION VALUE="Skin and Wrinkle Treatments">Skin and Wrinkle Treatment
<OPTION VALUE="Vericose and Spider Veins">Vericose and Spider Veins
</SELECT>
</td>
</tr>
</table>
</div>
<div id="comwrap">
<table celllpadding="5" border="0">
<tr>
<td>
   &nbsp;Comments: <br /> &nbsp;&nbsp;<textarea rows="3" name="message" cols="45"></textarea>&nbsp;&nbsp;&nbsp;<br /><br /></td>
</tr></table></div>
<table cellpadding="10">
<tr>
<td>
<?php require_once('recaptchalib.php');
$publickey = "6LfoJgsAAAAAAMsA-8ujJwc-9Ige3plEqJ0VuHbS";
echo recaptcha_get_html($publickey); ?>
</td>
<td>   
<center><input type="submit" value="Submit Details" name="submit"></center>
</td>
</tr>
</table>
</form>
</div>

<?php 

$to = "email goes here";
$subject = "New Lead";
$firstname_field = $_POST['firstname'];
$lastname_field = $_POST['lastname'];
$phone_field = $_POST['phone'];
$email_field = $_POST['email'];
$age_field = $_POST['age'];
$sex_field = $_POST['sex'];
$address_field = $_POST['addresss'];
$city_field = $_POST['city'];
$message = $_POST['message'];
$procedure_field = $_POST['procedure'];
$body = "First Name: $firstname_field\n Last Name: $lastname_field\n E-Mail: $email_field\n Phone Number: $phone_field\n Age: $age_field\n Sex: $sex_field\n Address: $address_field\n City: $city_field\n Interested In: $procedure_field\n Comments:\n $message";

require_once ('recaptchalib.php');
$privatekey = "6LfoJgsAAAAAAEJpEt8BDGOzM54rqIz2JxkqY0T8";
$resp = recaptcha_check_answer ($privatekey,
                                $_SERVER["REMOTE_ADDR"],
                                $_POST["recaptcha_challenge_field"],
                                $_POST["recaptcha_response_field"]);

if (!$resp->is_valid)
{
$result="The Captcha was Entered Incorrectly. Please Go Back and Try Again.";
}

else if ($resp->is_valid)
{
$result="Your Form Has Been Submitted. A doctor will be in touch soon!";
mail($to, $subject, $body);
}

echo $result;
 
?>