Possible to use CAPTCHA module on a custom form? (no Drupal Form API)

Hi,
Is it possible to use this module on a custom form (I am not using the Drupal Form API or webform module)? I have a D7 site that has a node/content page with the following code:

<form id="signup_form" action="[form action]" method="post" enctype="application/x-www-form-urlencoded">
<div>
    <input id="firstName" tabindex="1" type="text" name="firstName" placeholder="First Name" value="" maxlength="100" class="name-icon" required />
    <label for="firstName">First Name* </label>
</div>

<div>
    <input id="lastName" tabindex="2" type="text" name="lastName" placeholder="Last Name" value="" maxlength="100" class="name-icon" required />
    <label for="lastName">Last Name* </label> 
</div>

<div> 
    <input id="email" tabindex="3" type="text" name="email" placeholder="Email" value="" maxlength="254" class="email-icon" />
    <label for="email">Email* </label>
</div>

<div>
    <input id="name" tabindex="4" type="text" name="name" placeholder="Company Name" value="" maxlength="100" class="company-icon" />
    <label for="name">Company Name* </label>
</div>

<div>
    <input id="phone" tabindex="5" type="text" name="phone" placeholder="Phone" value="" maxlength="40" class="phone-icon" />
    <label for="phone">Phone </label>
</div>

<input type="hidden" name="form_id" value="signup_form" />
<div>
    <input id="btn-submit" tabindex="6" class="submit" type="submit" name="submitButton" value="SIGN UP" />
</div>

</form>

I received a security requirement on our sign up form (Drupal 7 site), to have reCAPTCHA displayed at the bottom of the page when there has been an X number of form submissions from the same IP address in a given amount of time.   I was hoping CAPTCHA After could work for my custom form -- I tested it on our site's Admin Login page (/user) and it worked great, of course.

I can't seem to get any CAPTCHA displaying on my custom form page. In the General Settings, I selected "Add CAPTCHA administration links to forms" and a drop-down with "CAPTCHA: CHALLENGE "RECAPTCHA" ENABLED" displayed in the content page where the form is coded.

I just learned that the HTML id has nothing to do with $form_id... I referenced this Drupal Stack Exchange page, and haven't been able to generate the form ID...I'm wondering maybe my custom HTML form doesn't have a $form_id because it doesn't use any drupal form modules. If that's the case, then I probably can't use the CAPTCHA module. Could someone please confirm?

Any guidance would be great. Thank you!