How to use word 2010 to write and edit equations in a doc file

No, that is not possible. When you save as a doc file, you are telling Word that you want the document to contain only those features that were available in versions of Word up to 2003. Because the equation editor you used was first introduced in Word 2007, the equations are converted to pictures.

However, if you don't try to edit those equations in Word 2003 or earlier, and you later reopen the same document in Word 2007 or later, you can use the Convert button on the File menu (or the Office button in 2007), and the equations will be restored to their editable form. Of course, you'd then have to save it in docx format to keep the equations editable.

The other alternative is to write all your equations in the old Equation Editor 3.0, which is still included in Office 2010. It may not be installed by default, so you may have to open the Office installer, choose to add features, and select Equation Editor under the Office Tools folder. To use it, you can click Insert > Object and select "Microsoft Equation 3.0" in the list on the Create New tab.

Unfortunately, there is no automatic way to convert existing equations from one editor's format to the other.