Why didn't GM use a V8 in the H3

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Could be due to many reasons:
Development costs for the vehicle startup, and eventually they might put one in. Manufacturer allocates a set amount to design, test, and startup a vehicle. Following years may allow for another engine package to be developed tested, and released for production.
Possibly a small V8 was designated, tested and approved; however, the engine plant producing this engine may have been at max capacity, and ramp up time would be six months to a year, or startup another factory to build the same engine.
Packaging of a V8 under the hood; don't think we will ever see a large V8 block in that little compartment, but a small 5.3 V8 might fit.
Could have been some problems with crash tests, all different powertrain variations of a vehicle have to pass all crash tests. GM might have failed with a V8 and are working on a redesign to pass the tests.
Could be they didn't want to install one; period.
Many vehicles are initially released with one engine, and later in the model year, or the following model year different engine combinations are released.
The new solstice is released with a 4-cylinder, with rumors about a turbo coming in six months.
Saturn ION coupe was released with a 4-cylinder, following year a supercharged version was released. Just two examples.
What the actual reason was for the release of only this engine may never be known, unless the person(s) who made the decision step forward. But it can be fun to speculate.