How do I use inline styling with WordPress?

You will have to specify template directory in your inline background style too.
Use it like this.

<header class="entry-header" style="background: url('<?php bloginfo('stylesheet_directory'); ?>/images/bg_the-firm.jpg')no-repeat left top;">

If WordPress strip your inline styles, you can use body classes to add specific styles to each page.

Each page have some body classes, you can find those in page source in any browser. And define your header background like this in your style.css.

body.page-id-6 header.entry-header{background: url('images/bg_the-firm.jpg') no-repeat left top;}

Make sure you define page specific styles at the end of style.css to avoid overwriting it’s effect.