Does that page only appear when you use your Firefox desktop/taskbar shortcut to start Firefox, or does it appear in any new window (Ctrl+n)?

Any new window: Check your startup/home page setting here:

• Startup, home page, tabs, and download settings
• How to set the home page

If that is not changeable or reverts when you exit/restart Firefox, then we'll have you look for possible malware or optional preference file issues.

Only the shortcut: Check that the "target" of the shortcut has not been hacked. right-click the shortcut > Properties > Shortcut tab

The Target should be no more and no less than this:

"C:\Program Files (x86)\Mozilla Firefox\firefox.exe"

I looked up Dregol search and there may well be external components you need to remove to prevent it from being reinserted into your browsers. You may already have done a first pass, but here are some additional suggestions:

(1) Open the Windows Control Panel, Uninstall a Program. After the list loads, click the "Installed on" column heading to group the infections, I mean, additions, by date. This can surface undisclosed bundle items that otherwise are hard to spot in alphabetical order. Take out as much trash as possible here.

(2) You can search for remaining issues with the scanning/cleaning tools listed in our support article: Troubleshoot Firefox issues caused by malware. These on-demand scanners are free and take considerable time to run. If they finish quickly and especially if they require payment, you may have a serious infection. I suggest the specialized forums listed in the article in that case.

Finally, you can delete the Dregol search plugin using the Search panel in Options:

"3-bar" menu button (or Tools menu) > Options > Search