You'd think that conditional formatting would provide this capability, but it does not allow you to reference other cells. (Unlike Excel.)
Here is a script that will do the job. (If you're new to Google Apps Script, read this introduction.)
/**
* The onEdit() function, when defined, is automatically invoked whenever the
* spreadsheet is edited. (Not every change is an "edit".)
*
* If all of the "fill in the blanks" columns are empty, set the
* background color of the name column in that row.
*/
function onEdit(event) {
var row = event.range.getRow();
var sheet = event.range.getSheet();
var cellA = sheet.getRange(row,1);
// Get the range of cells B-J that make up the "fill in the blanks" range...
var fillInTheBlanks = sheet.getRange(row, 2, 1, 9)
.getValues()[0] // ... get values in 0-th row
.join(''); // ... and join into one string
// Check that string - if it is empty, the range was blank,
// and we should set the background color to, say, yellow.
if (fillInTheBlanks.length == 0) {
cellA.setBackground("yellow");
}
// On the other hand, if the string was NOT blank, there is something
// in the "fill in the blanks" range, so we will clear the background.
else {
cellA.setBackground("white");
}
}
/**
* The onOpen() function, when defined, is automatically invoked whenever the
* spreadsheet is opened.
*
* Iterates through all rows in spreadsheet, faking onEdit triggers, as if
* each name has been edited.
*/
function onOpen() {
var sheet = SpreadsheetApp.getActiveSpreadsheet();
var lastRow = sheet.getLastRow();
for (var row=1; row <= lastRow; row++) {
onEdit({range:sheet.getRange('A'+row)});
}
};