The code is public domain.

<?php
    error_reporting(E_ALL ^ E_DEPRECATED);

    /* Settings */
    $size = 15;                  /* font size */
    $position = 0.5;             /* spacing */
    $psize = $size*0.6;         /* upper index size */
    $font = __DIR__ . "/7.ttf";         
    $maxpower = 3;                 /* maximum power of polynomial */
    $minvars = 2;                 /* minimum number of variables */
    $maxvars = 3;                 /* maximum number of variables */

    /* Randomize */
    list($usec, $sec) = explode(' ', microtime());
    $mt = (float) $sec + ((float) $usec * 100000);
    srand($mt);

    /* Generate polynomial */
    $letters = array();
    for ($i=ord("a");$i<=ord("z");$i++) {
        $letter = chr($i);
        if ($letter == "o" || $letter == "l" || $letter == "e" || $letter == "i") { continue; }
        $letters[] = $letter;
    }
    $parts = array();
    $cnt = rand($minvars,$maxvars);
    for ($i=0;$i<$cnt;$i++) { /* all variables */
        $idx = rand(0,count($letters)-1);
        $letter = $letters[$idx];
        
        for ($j=$maxpower;$j>=1;$j--) { /* all powers */
            if (rand(1,2) == 1) { continue; }
            $modifier = rand(1,9);
            if (rand(1,2) == 1) { $modifier *= -1; }
            $parts[] = array("letter"=>$letter,"modifier"=>$modifier,"power"=>$j);
        }
        
        unset($letters[$idx]);
        $new = array();
        foreach ($letters as $l) { $new[] = $l; }
        $letters = $new;
    }
    
    $power = (count($parts) ? $parts[0]["power"] : 1);
    $solution = (count($parts) ? $parts[0]["modifier"] : 0);
    $variable = (count($parts) ? $parts[0]["letter"] : $letters[rand(0,count($letters)-1)]);
    
    for ($i=1;$i<=$power;$i++) { $solution *= $i; } /* adjust solution by product of powers */
    shuffle($parts);
    
    if (!count($parts) || rand(1,2)==1) { /* constant part */
        $modifier = rand(1,9);
        if (rand(1,2) == 1) { $modifier *= -1; }
        $parts[] = array("letter"=>"", "modifier"=>$modifier, "power"=>1);
    }
    
    /* Initialize image */
    $captcha = imageCreate(1000,50);
    $background = imageColorAllocate($captcha, 235, 235, 255);
    $black = imageColorAllocate($captcha, 0, 0, 0);
//    imageColorTransparent($captcha, $background);

    /* Draw formula */
    if (rand(1,2) == 1) { /* minus before formula */
        $solution *= -1;
        imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, "–");
        $position += 0.7;
    }
    
    /* Partials */
    imagettftext($captcha, $size, 0, (($position+0.25)*25), 17, $black, $font, "∂");
    if ($power != 1) { 
        imagettftext($captcha, $psize, 0, (($position+0.7)*25), 10, $black, $font, $power);
    }
    imagettftext($captcha, $size, 0, (($position)*25), 30, $black, $font, "―");
    imagettftext($captcha, $size, 0, (($position+0.4)*25), 30, $black, $font, "―"); /* hack for monospace */
    imagettftext($captcha, $size, 0, (($position+0.65)*25), 30, $black, $font, "―"); /* hack for monospace */
    imagettftext($captcha, $size, 0, ($position*25), 43, $black, $font, "∂");
    $position += 0.5;
    imagettftext($captcha, $size, 0, ($position*25), 43, $black, $font, $variable);
    if ($power != 1) { 
        imagettftext($captcha, $psize, 0, (($position+0.5)*25), 36, $black, $font, $power);
    }
    $position += 0.7;
    
    imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, "(");
    $position += 0.5;
    for ($i=0;$i<count($parts);$i++) {
        $p = $parts[$i];
        $modifier = $p["modifier"];
        $letter = $p["letter"];
        $power = $p["power"];
        $pre = false;
        if ($i || $modifier < 0) { $pre = ($modifier > 0 ? "+" : "–"); }
        if ($pre) {
            imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, $pre);
            $position += 0.75;
        }
        if (abs($modifier) != 1 || !$letter) {
            imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, abs($modifier));
            $position += 0.6;
        }
        imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, $letter);
        if ($power != 1) {
            imagettftext($captcha, $psize, 0, (($position+0.5)*25), 20, $black, $font, $power);
            $position += 0.1;
        }
        if ($letter) { $position += 0.75; }
        
    }
    /* End */
    imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, ")");
    $position++;
    imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, "=");
    $position++;
    imagettftext($captcha, $size, 0, ($position*25), 30, $black, $font, $solution);
    $position+=strlen($solution)/2 + 0.5;
    
    $c_old = $captcha;
    $captcha = imagecreate($position*25, imagesy($c_old));
    imagecopy($captcha, $c_old, 0, 0, 0, 0, imagesx($captcha), imagesy($captcha));

    /* Cache control */
    header("Expires: Wed, 1 Jan 1997 00:00:00 GMT");
    header("Last-Modified: ".gmdate("D, d M Y H:i:s")." GMT");
    header("Cache-Control: no-store, no-cache, must-revalidate");
    header("Cache-Control: post-check=0, pre-check=0", false);
    header("Pragma: no-cache");

    /* Display the image */
    header('Content-type: image/gif');
    imageGif($captcha);
    imageDestroy($captcha);
?>