Whitening toothpaste is a maintenance tool. It removes extrinsic stains as they form and, in some formulations, adds a very low dose of peroxide. Expect a subtle shift in brightness over weeks of use.
Whitening strips are a treatment. They hold hydrogen peroxide against enamel for a fixed session (typically 30 minutes) and, over a 10-20 day course, produce a measurable intrinsic shade change.
The right answer is usually both: use strips (or trays) to change your baseline, then use a whitening toothpaste daily to keep it there.
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