Sum Cells Multiplied by Offsetted Cells

[Smally]

In the attached spreadsheet. The yellow cells multiply the cells below with the cells in column B per row and adds them all together.

Is it possible to do this using ranges/formulas as multiplying and adding them individually can lead to accidently missing rows out.
The actual spreadsheet has hundreds of rows, with random gaps, and sometimes users insert a row and don't modify the yellow cells to include them.

Unfortunately, I cannot add columns or hide them.