Hello, I'm a novice with flooring. My oak floors were stained dark by the previous owner. I cut into a wall to add a bathroom which turned out great, but it exposed a strip of flooring that is not stained dark. My wife wants it to match. How do I stain and finish just the strip? Do I need to sand off whatever is on the light wood strip so that a stain will take? What do I seal it with in the end?

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You will need to have bare wood to get any stain absorbing into the wood's surface. A dark stain is easier to match if you dampen the surface with water and let it dry, popping the grain, which allows the cells to open so more stain soaks in.
Try sanding a sample piece that you prepare like the flooring so you have an idea of the color of the sample on wood before you start to stain the floor. The color you will get after the stain has dried thoroughly and is sealed with the first coat of finish will be very similar to the stain color when it is still wet on the sample or on the flooring. You can use shellac or regular poly-urethane for the seal coat. Then light sanding and finish of the gloss level of your floor. Usually floors are finished with "satin" finish.