display recaptcha

Good morning
I have a wordpress custom form. I would like to add your recaptcha. I added a shortcode like: <?php echo do_shortcode(‘[captchac cap53][captchar cap53]’);?> or by directly adding the following code before my “submit” html tag:

<?php  $captcha_instance = new ReallySimpleCaptcha( );
$word = $captcha_instance->generate_random_word();
$prefix = mt_rand();
$captcha_instance->generate_image( $prefix, $word );
$correct = $captcha_instance->check( $prefix, $the_answer_from_respondent );
$captcha_instance->remove($prefix); ?>

But nothing is displayed. The PHP modules on the server are well installed. Did I make a mistake, an oversight? How to integrate in a custom form?
THANKS
Nicholas

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